*315*
*72*
*10MB*

*English*
*Pages 1109*
*Year 2008*

- Author / Uploaded
- Ronald J. Harshbarger
- James J. Reynolds

- Similar Topics
- Economy
- Mathematical Economics

Index of Selected Applications Management Accounting, 230 Advertising, 8, 27, 49, 84, 106, 291, 323, 324, 367, 382, 384, 394, 398, 399, 496, 505, 509, 526, 576, 600, 613, 651, 659, 666, 672, 690, 695, 703, 709, 722, 723, 742, 743, 794, 834, 844, 892, 903, 962, 965, 999, 1002 Agricultural business, 287, 292, 304, 307, 319, 326, 329, 345, 353, 401 Airlines, 136, 192 Annual reports, 214 Average cost, 49, 54, 709, 724, 727, 760, 762, 763, 824, 900, 949 Beneﬁts, 540, 989 Break-even analysis, 69, 121, 127, 164, 167, 197, 200 Budgeting, 230, 323 Business risk, 203 Capital investment, 642, 760, 790, 800 Cellular phones, 35, 36, 106, 193, 399, 533, 580, 643, 666, 668, 672, 745, 825 Committees, 508, 511, 529 Company growth, 27, 400, 672, 737 Competition, 258 Competition in telecommunications, 533 Construction, 305, 323, 325, 459 Cost, 83, 84, 119, 126, 133, 135, 136, 151, 160, 169, 176, 181, 182, 183, 199, 212, 275, 335, 392, 393, 615, 629, 642, 676, 686, 687, 688, 689, 813, 845, 846, 849, 852, 866, 903, 920, 927, 943, 949, 956, 971, 997, 999, 1000 Cost-beneﬁt, 54, 83, 96, 100, 107, 197, 595, 601, 613, 642, 650, 755, 756 Customer service, 576, 894, 936 Decision making, 477, 516 Diminishing returns, 711, 715, 722, 723, 760 Egg production, 228 Electrical power use, 182, 614, 951 Employee age and gender, 475, 476, 481, 484, 486 Employee evaluation, 517 Expense accounts, 218 Franchise growth, 835 Fundraising, 138 Hiring, 496, 511, 529 Hospital administration, 137 Hotel pricing, 765 Human resources, 576 Industrial consolidation, 367 Insurance, 424, 446, 517, 540, 552, 553, 556, 563, 572, 574, 576 Interdepartmental costs, 270, 271 Inventory, 249, 476, 571, 737, 740, 744, 761, 763, 903, 986 Job bidding, 486 Job effectiveness, 53 Joint cost, 972, 977 Labor efﬁciency, 219 Loss, 54 Maintenance costs, 948 Management, 218, 219, 290, 305, 475, 476, 477, 511, 516, 529, 539 Manufacturing, 84, 259, 267, 275, 286, 290, 291, 302, 303, 304, 305, 314, 322, 323, 324, 325, 333, 344, 345, 348, 349, 935, 989, 998 Marginal cost, 120, 126, 135, 136, 673, 675, 676, 681, 772, 775, 783, 812, 963, 972, 977 Marginal proﬁt, 120, 126, 135, 678, 679, 681, 682, 689, 760 Marginal return to sales, 690, 965

Marginal revenue, 615, 630, 631, 647, 674, 680, 681, 689, 709, 775, 780, 783 Market share, 372, 394, 579, 762, 866 Marketing, 476 Minimizing average cost, 727, 733, 734 Minimizing costs, 333, 334, 335, 344, 345, 348, 349, 351, 691, 738, 744, 745, 756, 760, 763, 764 Oil reﬁneries, 229 Operating leverage, 203 Organizational growth, 388, 393, 394 Package design, 36, 84, 181, 739, 763, 764 Parking costs, 601 Parts delivery, 485 Parts listings, 267, 272 Postal rates, 182, 614 Pricing, 90, 117, 118, 228, 658, 709, 765, 862, 1003 Pricing for maximum proﬁt, 57 Printing, 761 Product design, 761 Product display, 511 Product reliability, 367, 392, 551, 834, 890, 894 Production, 219, 228, 230, 235, 272, 274, 275, 344, 349, 665, 709, 723, 790, 795, 834, 868, 872, 893, 944, 955, 958, 961, 971, 973, 978, 988, 989, 994, 997, 999, 1000, 1001 Production costs, 220, 222, 277, 298, 304, 348, 709, 743, 750, 764 Productivity, 528, 600, 612, 630, 642, 665, 687, 709, 722, 743, 756, 760, 761, 834, 866 Proﬁt, 35, 54, 64, 70, 83, 84, 119, 121, 126, 134, 135, 136, 149, 150, 159, 196, 244, 280, 393, 411, 600, 630, 678, 679, 687, 688, 711, 715, 743, 756, 800, 813, 866, 867, 903, 961, 971, 988, 989, 1000, 1001 Proﬁt maximization, 57, 154, 166, 168, 197, 200, 292, 303, 305, 306, 307, 314, 323, 324, 325, 336, 340, 344, 348, 349, 350, 351, 353, 426, 683, 686, 729, 734, 753, 760, 761, 762, 765, 848, 849, 852, 979, 983 Project teams, 509 Property development, 482, 741, 945 Publicity, 576 Purchasing, 70, 329 Purchasing electric power, 951 Quality control, 472, 476, 477, 491, 495, 496, 499, 513, 516, 517, 528, 529, 538, 539, 540, 563, 564, 571, 576, 579, 936, 948, 949 Rentals, 35, 70, 117, 118, 160, 244, 447, 475, 484, 496, 530 Revenue, 35, 42, 54, 84, 85, 119, 126, 135, 158, 160, 199, 228, 387, 393, 562, 576, 579, 600, 620, 629, 639, 641, 642, 650, 651, 658, 661, 663, 665, 671, 672, 674, 686, 687, 688, 689, 710, 743, 745, 756, 760, 812, 813, 818, 821, 824, 831, 834, 843, 849, 866, 949, 997 Revenue maximization, 156, 197, 295, 325, 345, 724, 725, 733, 761, 763, 765 Rewards for employees, 515 Safety, 192, 478, 801, 812 Salaries, 70, 411, 412, 459, 484, 551, 552 Sales, 217, 456, 511, 516, 552, 563, 600, 612, 613, 756, 862 Sales decay, 382, 384, 391, 392, 398, 399, 782, 813 Sales growth, 390, 393, 394, 398 Sales promotion, 475 Scheduling, 290, 303, 304, 323, 333, 334, 335, 336, 340, 344 Shadow prices, 304, 319, 353 Shipping, 305, 323 Starbucks stores, 401

Telecommunications, 399, 581, 881, 893, 915, 918 Ticket sales, 244 Time study, 49 Transportation, 27, 245, 252, 258, 259, 276, 352 Unions, 161, 218, 666 Utilities, 94, 95, 178, 182, 197, 529, 601, 625 Warranties, 936 Wilson’s lot size formula, 960, 971 Wireless service spending, 71, 882, 893 Economics Balance of trade, 206, 210, 217 Budget deﬁcit, 193 Cobb-Douglas production function, 955, 958, 961, 974, 978, 994, 995, 997, 1001 Competitive market, 975 Competitive products, 976, 1003 Complementary products, 976 Consumer expenditure, 41, 106, 117, 630, 743, 1001 Consumer price index, 36, 95, 188, 364, 368, 710, 723, 785, 825, 845 Consumer’s surplus, 908, 909, 912, 914, 915, 920, 948 Continuous income streams, 883, 889, 893, 904, 905, 913, 914, 920, 921, 924, 927, 943, 948, 949 Demand, 122, 128, 135, 136, 161, 162, 163, 167, 182, 197, 199, 387, 392, 393, 569, 612, 629, 642, 653, 657, 659, 665, 687, 776, 785, 792, 795, 800, 813, 869, 903, 943, 944, 975, 979, 1001 Dow Jones averages, 7, 19, 73, 82, 601, 736 Economy models, 260, 266, 276, 278 Elasticity of demand, 802, 803, 804, 805, 809, 812, 813 Employment, 496 Gasoline mileage and prices, 552, 571 Gini coefﬁcient of income, 895, 899, 904, 927, 944, 950 Gompertz curves, 382, 393, 398, 863 Gross domestic product, 182, 192, 643, 660, 724, 863 Income distribution, 894, 899, 904, 927, 944, 948, 950 Indifference curves, 958, 961 Inﬂation, 362, 392, 397, 426, 812, 814, 836, 837, 864 Job growth, 7 Leontief input-output models, 260, 261, 262, 263, 264, 266, 269, 270, 271, 272, 276, 278 Lorenz curves, 894, 904, 927, 944, 950 Marginal demand, 976, 1000 Marginal productivity, 973, 974 Marginal return to sales, 690, 965 Marginal utility, 971 Market equilibrium, 123, 124, 128, 135, 161, 162, 163, 167, 197, 199 Market equilibrium after taxes, 125, 130 Monopoly market, 910 National consumption, 665, 850, 851, 853, 866, 867 National debt, 368, 392, 614, 763, 783 Oil imports, 881, 894, 950 Open and closed economies, 264, 266, 269, 271, 278 Optimization in business, 691 Personal income, 14, 19, 368, 399, 631, 637, 784, 825, 845 Producer’s surplus, 911, 912, 915, 920, 927, 948, 949 Purchasing power, 362, 364, 392, 393, 784, 813 Social Security beneﬁts, 82, 141, 146, 149, 399 Supply, 122, 128, 135, 136, 161, 162, 163, 167, 197, 199, 392, 393, 687, 775, 800, 869, 944 Taxation, 15, 70, 71, 73, 76, 94, 107, 125, 130, 150, 182, 183, 186, 193, 279, 601, 602, 606, 613, 660, 761, 808, 810, 812, 813, 904, 1000 Tourism spending, 150, 181, 399, 635, 784, 882, 927

Unemployment, 550 Utility, 958, 961, 971, 990, 993, 997, 999, 1000, 1001 Work force, 83, 101, 106, 161, 207, 381, 485, 616, 652, 686, 724, 756, 776, 815, 835 Finance Amortization, 449, 450, 454, 455, 456, 457, 458, 460, 461, 959 Annuities, 49, 427, 428, 429, 430, 431, 434, 435, 436, 437, 438, 439, 440, 445, 446, 447, 448, 459, 460, 613 Annuities due, 433, 435, 442, 443, 446 APY, 418, 424, 459, 461 ATM transactions, 515 Banks, 85, 91, 305, 511, 516 Bond pricing, 440 Bonds, 441, 446, 447, 460, 461, 462, 552 Budgeting, 323, 563 Capital value, 929, 935, 949 College savings, 412, 414, 418, 423, 424, 435, 436, 448, 459, 461, 462 Compound interest, 19, 33, 367, 368, 393, 398, 412, 414, 415, 417, 423, 458, 459, 460, 461, 462, 777, 779, 811, 812, 844, 861 Consumer credit, 397 Court settlements, 443, 447, 461 CPA exam, 564 Credit cards, 69, 517, 518 Debt, 218, 450, 454, 455, 458, 461 Debt reﬁnancing, 82 Deferred annuities, 444, 448 Delinquent accounts, 486, 490 Depreciation, 69, 86, 94, 95, 136, 426, 892 Doubling time, 371, 381, 419 Earnings, 660, 709 Future value, 53, 458 Future value of annuities, 427, 428, 429, 430, 431, 434, 435, 436, 459 Future value of income stream, 907, 914 Future value of investments, 33, 53, 62, 357, 359, 367, 393, 398, 410, 414, 423, 424, 425, 459, 460, 782, 959 Home equity loans, 463 Income levels, 529 Income stream, 913, 914, 920, 921, 924, 927, 943, 948, 949 Investing, 26, 33, 35, 41, 62, 65, 70, 110, 118, 134, 136, 239, 244, 245, 246, 259, 275, 277, 278, 325, 357, 359, 367, 368, 369, 371, 381, 393, 398, 404, 405, 410, 415, 417, 419, 423, 424, 425, 435, 436, 446, 459, 460, 462, 659, 775, 862, 866, 959 Loans, 53, 82, 118, 244, 410, 411, 450, 453, 454, 455, 456, 457, 458, 459, 460, 463, 1001 Mortgages, 76, 81, 133, 449, 450, 456, 457, 462, 464, 960, 970 Mutual funds, 394, 516 Net worth, 579 Perpetuities, 613 Personal income, 19 Present value, 19, 410, 458, 906, 914, 927, 943 Present value of annuities, 437, 438, 445, 446, 448 Purchasing a home, 464 Real estate, 117, 551, 836, 837 Retirement planning, 95, 160, 424, 425, 435, 436, 446, 447, 448, 461, 462, 951, 989, 999 Savings, 410, 425, 435, 436, 448, 459, 460, 948, 999 Simple interest, 41, 404, 405, 410, 411, 458, 459, 460, 461 Sinking funds, 431, 435, 436, 459 Stock market, 5, 7, 19, 368, 393, 410, 415, 424, 425, 460, 529, 601, 813 Taxes, 15, 70, 71, 73, 94, 107, 124, 130, 279, 280, 601, 602, 606, 613, 660, 904

NINTH EDITION

Mathematical Applications for the Management, Life, and Social Sciences

This page intentionally left blank

NINTH EDITION

Mathematical Applications for the Management, Life, and Social Sciences Ronald J. Harshbarger University of South Carolina Beaufort

James J. Reynolds Clarion University of Pennsylvania University of South Carolina Beaufort

Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States

Senior Sponsoring Editor: Cathy Cantin Associate Editor: Jeannine Lawless Media Editor: Catie Ronquillo Marketing Specialist: Ashley Pickering Marketing Communications Manager: Mary Anne Payumo Project Manager, Editorial Production: Shelley Dickerson Art and Design Manager: Jill Haber

Senior Content Manager: Maren Kunert Manufacturing Buyer: Diane Gibbons Permissions Editor: Craig Mertens Text Designer: Jean Hammond Photo Manager: Jennifer Meyer Dare Photo Researcher: Stacey Dong Cover Designer: Anne Katzeff Cover Image: © Stuttgart Pforzheim/Getty Images Compositor: Aptara®, Inc.

Photo Credits: p. 1, Gail Shumway/Getty Images; p. 58, © Natalie Behring/Digital Railroad; p. 139, © Patrick Brice/Arcaid/Corbis; p. 201, NASA; p. 204, © Klaus Leidorf/zefa/Corbis; p. 281, © Wang Hoof/epa/Corbis; p. 355, © Roy Botterell/Corbis; p. 402, Ted Russell/Getty Images; p. 465, © Mike Denton/Alamy; p. 534, © Darell Gulin/CORBIS; p. 584, Al Bello/Getty Images; p. 693, © Nakheel Corporation; p. 766, © Pat Bennett/Alamy; p. 816, © John Edward Linden/Arcaid/Corbis; p. 870, © 2008 Jupiterimages Corporation; p. 953, © PCL/Alamy © 2009 Brooks Cole, Cengage Learning ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 or the 1976 United States Copyright Act, without the prior written permission of the publisher. For product information and technology assistance, contact us at Cengage Learning Academic Resource Center, 1-800-423-0563 For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions. Further permissions questions can be e-mailed to [email protected] Library of Congress Control Number: 2008921465 ISBN-13: 978-0-547-14509-9 ISBN-10: 0-547-14509-8 Brooks/Cole 10 Davis Drive Belmont, CA 94002 USA Cengage Learning products are represented in Canada by Nelson Education, Ltd. For your course and learning solutions, visit academic.cengage.com. Purchase any of our products at your local college store or at our preferred online store www.ichapters.com. Printed in the United States of America 1 2 3 4 5 6 7 12 11 10 09 08

Contents Preface

0

Algebraic Concepts

xii

1

0.1 Sets 2 0.2 The Real Numbers 9 0.3 Integral Exponents 15 0.4 Radicals and Rational Exponents 20 0.5 Operations with Algebraic Expressions 27 0.6 Factoring 37 0.7 Algebraic Fractions 42 Key Terms and Formulas 49 Review Exercises 51 Chapter Test 54 Extended Applications & Group Projects 56 Campaign Management • Pricing for Maximum Profit

1

Linear Equations and Functions

58

Warm-up 59 1.1 Solution of Linear Equations and Inequalities in One Variable 60 1.2 Functions 71 1.3 Linear Functions 85 1.4 Graphs and Graphing Utilities 96 Graphical Solutions of Equations 1.5 Solutions of Systems of Linear Equations 107 Three Equations in Three Variables 1.6 Applications of Functions in Business and Economics 119 Total Cost, Total Revenue, and Profit • Break-Even Analysis • Supply, Demand, and Market Equilibrium Key Terms and Formulas 130 Review Exercises 132 Chapter Test 135 Extended Applications & Group Projects 137 Hospital Administration • Fundraising

v

vi

● Contents

2

Quadratic and Other Special Functions

139

Warm-up 140 2.1 Quadratic Equations 141 Factoring Methods • The Quadratic Formula 2.2 Quadratic Functions: Parabolas 151 2.3 Business Applications of Quadratic Functions 161 Supply, Demand, and Market Equilibrium • Break-Even Points and Maximization 2.4 Special Functions and Their Graphs 169 Polynomial and Rational Functions • Piecewise Defined Functions 2.5 Modeling; Fitting Curves to Data with Graphing Utilities (optional) 183 Key Terms and Formulas 194 Review Exercises 195 Chapter Test 198 Extended Applications & Group Projects 201 An Inconvenient Truth • Body-Mass Index • Operating Leverage and Business Risk

3

Matrices

204 Warm-up 205 3.1 Matrices 206 3.2 Multiplication of Matrices 220 3.3 Gauss-Jordan Elimination: Solving Systems of Equations 230 Systems with Unique Solutions • Systems with Nonunique Solutions 3.4 Inverse of a Square Matrix; Matrix Equations 246 Matrix Equations • Determinants 3.5 Applications of Matrices: Leontief Input-Output Models 260 Key Terms and Formulas 273 Review Exercises 273 Chapter Test 276 Extended Applications & Group Projects 279 Taxation • Company Profits After Bonuses and Taxes

4

Inequalities and Linear Programming

281

Warm-up 282 4.1 Linear Inequalities in Two Variables 283 4.2 Linear Programming: Graphical Methods 292 4.3 The Simplex Method: Maximization 306 Nonunique Solutions: Multiple Solutions and No Solution • Shadow Prices 4.4 The Simplex Method: Duality and Minimization 326 4.5 The Simplex Method with Mixed Constraints 336

Contents ●

vii

Key Terms 345 Review Exercises 346 Chapter Test 350 Extended Applications & Group Projects 352 Transportation • Slack Variables and Shadow Prices

5

Exponential and Logarithmic Functions

355

Warm-up 356 5.1 Exponential Functions 357 Modeling with Exponential Functions 5.2 Logarithmic Functions and Their Properties 369 Logarithmic Functions and Graphs • Modeling with Logarithmic Functions • Properties of Logarithms • Change of Base 5.3 Solution of Exponential Equations: Applications of Exponential and Logarithmic Functions 382 Solving Exponential Equations Using Logarithmic Properties • Growth and Decay • Economic and Management Applications • Gompertz Curves and Logistic Functions Key Terms and Formulas 395 Review Exercises 396 Chapter Test 398 Extended Applications & Group Projects 400 Starbucks Stores • Agricultural Business Management

6

Mathematics of Finance

402

Warm-up 403 6.1 Simple Interest; Sequences 404 Simple Interest • Sequences • Arithmetic Sequences 6.2 Compound Interest; Geometric Sequences 412 Compound Interest • Geometric Sequences 6.3 Future Value of Annuities 427 Ordinary Annuities • Annuities Due 6.4 Present Values of Annuities 437 Ordinary Annuities • Annuities Due • Deferred Annuities 6.5 Loans and Amortization 449 Unpaid Balance of a Loan Key Terms and Formulas 457 Review Exercises 458 Chapter Test 461 Extended Applications & Group Projects 463 Mail Solicitation Home Equity Loan: Is This a Good Deal? • Profit Reinvestment • Purchasing a Home

viii

● Contents

7

Introduction to Probability 465 Warm-up 466 7.1 Probability; Odds 467 7.2 Unions and Intersections of Events: One-Trial Experiments 478 7.3 Conditional Probability: The Product Rule 486 7.4 Probability Trees and Bayes’ Formula 497 Probability Trees • Bayes’ Formula 7.5 Counting: Permutations and Combinations 505 7.6 Permutations, Combinations, and Probability 512 7.7 Markov Chains 517 Key Terms and Formulas 526 Review Exercises 527 Chapter Test 530 Extended Applications & Group Projects 532 Phone Numbers • Competition in the Telecommunications Industry

8

Further Topics in Probability; Data Description

534

Warm-up 535 8.1 Binomial Probability Experiments 536 8.2 Data Description 540 8.3 Discrete Probability Distributions; The Binomial Distribution 553 Discrete Probability Distributions • Measures of Dispersion • The Binomial Distribution 8.4 Normal Probability Distribution 564 8.5 The Normal Curve Approximation to the Binomial Distribution 572 Key Terms and Formulas 577 Review Exercises 578 Chapter Test 580 Extended Applications & Group Projects 582 Lotteries • Statistics in Medical Research; Hypothesis Testing

9

Derivatives

584 Warm-up 585 9.1 Limits 586 9.2 Continuous Functions; Limits at Infinity 602 Summary 9.3 Average and Instantaneous Rates of Change: The Derivative 9.4 Derivative Formulas 631 9.5 The Product Rule and the Quotient Rule 644 9.6 The Chain Rule and the Power Rule 653 9.7 Using Derivative Formulas 661

615

Contents ●

9.8 Higher-Order Derivatives 666 9.9 Applications of Derivatives in Business and Economics 673 Key Terms and Formulas 683 Review Exercises 684 Chapter Test 688 Extended Applications & Group Projects 690 Marginal Return to Sales • Tangent Lines and Optimization in Business and Economics

10 Applications of Derivatives

693

Warm-up 694 10.1 Relative Maxima and Minima: Curve Sketching 695 10.2 Concavity: Points of Inflection 711 Second-Derivative Test 10.3 Optimization in Business and Economics 724 Maximizing Revenue • Minimizing Average Cost • Maximizing Profit 10.4 Applications of Maxima and Minima 737 10.5 Rational Functions: More Curve Sketching 745 Asymptotes • More Curve Sketching Key Terms and Formulas 757 Review Exercises 758 Chapter Test 762 Extended Applications & Group Projects 764 Production Management • Room Pricing in the Off-Season

11 Derivatives Continued

766

Warm-up 767 11.1 Derivatives of Logarithmic Functions 768 11.2 Derivatives of Exponential Functions 777 11.3 Implicit Differentiation 785 11.4 Related Rates 795 11.5 Applications in Business and Economics 802 Elasticity of Demand • Taxation in a Competitive Market Key Terms and Formulas 810 Review Exercises 811 Chapter Test 813 Extended Applications & Group Projects 814 Inflation • Knowledge Workers

ix

x

● Contents

12 Indefinite Integrals

816

Warm-up 817 12.1 The Indefinite Integral 818 12.2 The Power Rule 826 12.3 Integrals Involving Exponential and Logarithmic Functions 836 12.4 Applications of the Indefinite Integral in Business and Economics 845 Total Cost and Profit • National Consumption and Savings 12.5 Differential Equations 854 Separable Differential Equations • Applications of Differential Equations Key Terms and Formulas 864 Review Exercises 865 Chapter Test 867 Extended Applications & Group Projects 868 Employee Production Rate • Supply and Demand

13 Definite Integrals: Techniques of Integration

870

Warm-up 871 13.1 Area Under a Curve 872 13.2 The Definite Integral: The Fundamental Theorem of Calculus 883 13.3 Area Between Two Curves 894 13.4 Applications of Definite Integrals in Business and Economics 904 Continuous Income Streams • Consumer’s Surplus • Producer’s Surplus 13.5 Using Tables of Integrals 915 13.6 Integration by Parts 921 13.7 Improper Integrals and Their Applications 927 13.8 Numerical Integration Methods: Trapezoidal Rule and Simpson’s Rule 936 Key Terms and Formulas 945 Review Exercises 947 Chapter Test 949 Extended Applications & Group Projects 951 Retirement Planning • Purchasing Electrical Power

Contents ●

14 Functions of Two or More Variables

xi

953

Warm-up 954 14.1 Functions of Two or More Variables 955 14.2 Partial Differentiation 962 First-Order Partial Derivatives • Higher-Order Partial Derivatives 14.3 Applications of Functions of Two Variables in Business and Economics 972 Joint Cost and Marginal Cost • Production Functions • Demand Functions 14.4 Maxima and Minima 979 Linear Regression 14.5 Maxima and Minima of Functions Subject to Constraints: Lagrange Multipliers 990 Key Terms and Formulas 998 Review Exercises 999 Chapter Test 1001 Extended Applications & Group Projects 1002 Advertising • Competitive Pricing

Appendix AP-1 Table I: Future Value of an Ordinary Annuity of $1 (sn0i) AP-1 Table II: Present Value of an Ordinary Annuity of $1 (an0i) AP-3 Table III: Areas Under the Standard Normal Curve AP-5 Answers A-1 Index I-1

Preface

T

o paraphrase English mathematician, philosopher, and educator Alfred North Whitehead, the purpose of education is not to ﬁll a vessel but to kindle a ﬁre. In particular, Whitehead encouraged students to be creative and imaginative in their learning and to continually form ideas into new and more exciting combinations. This desirable goal is not always an easy one to realize in mathematics with students whose primary interests are in areas other than mathematics. The purpose of this text, then, is to present mathematical skills and concepts, and to apply them to ideas that are important to students in the management, life, and social sciences. We hope that this look at the relevance of mathematical ideas to a broad range of ﬁelds will help inspire the imaginative thinking and excitement for learning that Whitehead spoke of. The applications included allow students to view mathematics in a practical setting relevant to their intended careers. Almost every chapter of this book includes a section or two devoted to the applications of mathematical topics, and every section contains a number of application examples and problems. An index of these applications on the front and back inside covers demonstrates the wide variety used in examples and exercises. Although intended for students who have completed two years of high school algebra or its equivalent, this text begins with a brief review of algebra which, if covered, will aid in preparing students for the work ahead.

Pedagogical Features In this new edition, we have incorporated many suggestions that reﬂect the needs and wishes of our users. Important pedagogical features that have characterized previous editions have been retained. They are as follows. Intuitive Viewpoint. The book is written from an intuitive viewpoint, with emphasis on concepts and problem solving rather than on mathematical theory. Yet each topic is carefully developed and explained, and examples illustrate the techniques involved. Flexibility. At different colleges and universities, the coverage and sequencing of topics may vary according to the purpose of the course and the nature of the student audience. To accommodate alternate approaches, the text has a great deal of ﬂexibility in the order in which topics may be presented and the degree to which they may be emphasized. Chapter Warm-ups. With the exception of Chapter 0, a Warm-up appears at the beginning of each chapter and invites students to test themselves on the skills needed for that chapter. The Warm-ups present several prerequisite problem types that are keyed to the appropriate sections in the upcoming chapter where those skills are needed. Students who have difﬁculty with any particular skill are directed to speciﬁc sections of the text for review. Instructors may also ﬁnd the Warm-ups useful in creating a course syllabus that includes an appropriate scope and sequence of topics. Application Previews. Each section begins with an Application Preview that establishes the context and direction for the concepts that will be presented. Each of these Previews contains an example that motivates the mathematics in the section and is then revisited in a completely worked Application Preview example appearing later in the section. Comprehensive Exercise Sets. The overall variety and grading of drill and application exercises offer problems for different skill levels, and there are enough challenging problems to stimulate students in thoughtful investigations. Many exercise sets contain critical thinking and thought-provoking multistep problems that extend students’ knowledge and skills.

xii

Preface ●

xiii

Applications. We have found that integrating applied topics into the discussions and exercises helps provide motivation within the sections and demonstrates the relevance of each topic. Numerous real-life application examples and exercises represent the applicability of the mathematics, and each application problem is identiﬁed, so the instructor or student can select applications that are of special interest. In addition, we have found that offering separate lessons on applied topics such as cost, revenue, and proﬁt functions brings the preceding mathematical discussions into clear, concise focus and provides a thread of continuity as mathematical sophistication increases. There are ten such sections in the book, and entire chapters devoted to linear programming and ﬁnancial applications. Of the more than 5,500 exercises in the book, over 2,000 are applied. Extended Applications and Group Projects. Each chapter ends with at least two of these 32 case studies, which further illustrate how mathematics can be used in business and personal decision making. In addition, many applications are cumulative in that solutions require students to combine the mathematical concepts and techniques they learned in some of the preceding chapters. Graphical, Numerical, and Symbolic Methods. A large number of real data and modeling applications are included in the examples and exercises throughout the text and are denoted by the header Modeling. Many sections include problems with functions that are modeled from real data and some problems that ask students to model functions from the data given. These problems are solved by using one or more graphical, numerical, or symbolic methods. Graphing Utilities and Spreadsheets. In the Ninth Edition, we have increased the overall presence of our treatment of graphing utilities and especially spreadsheets. More examples, applications, Technology Notes, Calculator Notes, and Spreadsheet Notes, denoted by the icon , are scattered throughout the text. After the introduction to Excel in Section 0.5 and to graphing calculators in Section 1.4, discussions of the use of technology are placed in subsections and examples in many sections, so that they can be emphasized or de-emphasized at the option of the instructor. The discussions of graphing utility technology highlight its most common features and uses, such as graphing, window setting, trace, zoom, Solver, tables, ﬁnding points of intersection, numerical derivatives, numerical integration, matrices, solving inequalities, and modeling (curve ﬁtting). While technology never replaces the mathematics, it does supplement and extend it by providing opportunities for generalization and alternative ways of understanding, doing, and checking. Some exercises that are better worked with the use of technology, including graphing calculators, computer programs, and computer spreadsheets, are highlighted with the technology icon. Of course, many additional exercises can beneﬁt from the use of technology, at the option of the instructor. Technology can be used to graph functions and to discuss the generalizations, applications, and implications of problems being studied. In addition, the basics of spreadsheet operations are introduced and opportunities to solve problems with spreadsheets are provided. The Ninth Edition now includes more Excel information speciﬁcally relating to solving equations, systems of equations, quadratic equations, matrices, linear programming, output comparisons of f(x), f⬘(x), and f ⬙(x), and the maxima and minima of functions subject to constraints. Excel is also a particularly useful problem-solving tool when studying the Mathematics of Finance in Chapter 6. A separate Excel Guide offering examples and user information is available for purchase. Please see the Resources for the Student section on page xv of this Preface for more information. Checkpoints. The Checkpoints ask questions and pose problems within each section’s discussion, allowing students to check their understanding of the skills and concepts under discussion before they proceed. Solutions to these Checkpoints appear before the section exercises. Objective Lists. Every section begins with a brief list of objectives that outline the goals of that section for the student.

xiv

● Preface

Procedure/Example and Property/Example Tables. Appearing throughout the text, these tables aid student understanding by giving step-by-step descriptions of important procedures and properties with illustrative examples worked out beside them. Boxed Information. All important information is boxed for easy reference, and key terms are highlighted in boldface. Key Terms and Formulas. At the end of each chapter, just before the Chapter Review Exercises, there is a section-by-section listing of that chapter’s key terms and formulas. This provides a well-organized core from which a student can build a review, both to consult while working the Review Exercises and to identify quickly any section needing additional study. Review Exercises and Chapter Tests. At the end of each chapter, a set of Review Exercises offers students extra practice on topics in that chapter. These Reviews cover each chapter’s topics primarily in their section order, but without section references, so that students get a true review but can readily ﬁnd a section for further review if difﬁculties occur. A Chapter Test follows each set of Review Exercises. All Chapter Tests provide a mixture of problems that do not directly mirror the order of topics found within the chapter. This organization of the Chapter Test ensures that students have a ﬁrm grasp of all material in the chapter.

Changes in the Ninth Edition In the Ninth Edition, we continue to offer a text characterized by complete and accurate pedagogy, mathematical precision, excellent exercise sets, numerous and varied applications, and student-friendly exposition. There are many changes in the mathematics, prose, and art. The more signiﬁcant ones are as follows. ■

A new section, “Normal Approximation to the Binomial,” which discusses binomial probability applications involving large numbers of trials, has been added.

■

Three new Extended Applications/Group Projects have been added, two in Chapter 2 and one in Chapter 5. The two added in Chapter 2, “An Inconvenient Truth” and “Body Mass Index,” address important health, environmental, and economic issues. The “Inconvenient Truth” project is based on and utilizes Al Gore’s Oscar-winning ﬁlm of the same title to investigate some of the mathematics behind Gore’s Nobel Peace Prize-winning work on global warming.

■

Extended Applications/Group Projects have been updated as appropriate. These include Hospital Administration in Chapter 1, Phone Area Code Numbers in Chapter 7, and Powerball Lotteries in Chapter 8.

■

Expanded applications of matrix inverses explore the sensitivity of solutions in applications of systems of linear equations and in Input-Output models.

■

Drill Exercises throughout the text have been revised and reorganized to improve their grading and variety.

■

Solution of exponential and logarithmic equations (in Chapter 5) boasts expanded exposition, additional examples, and more comprehensive drill exercises.

■

Bond Pricing has been added to Section 6.4, Present Values of Annuities. This new application utilizes calculations of simple interest and present values of both compound interest investments and ordinary annuities and combines these to determine market prices for bonds.

■

Exercise Sets for Mathematics of Finance in Chapter 6 have been reorganized and revised to provide adequate drills as well as to expand the sets of problems that include miscellaneous ﬁnance applications.

Preface ●

xv

■

The section reference labels have been removed from the Chapter Review Exercises throughout the text. This improves the effectiveness of these exercises because students now must establish a context for each problem as well as solve it.

■

Data-driven and modeling examples and exercises have been updated or replaced with current applications.

■

An increased number of optional Excel discussions have been inserted where appropriate.

Resources for the Student Student Solutions Manual. Need a leg up on your homework or help to prepare for an exam? The Student Solutions Manual contains worked-out solutions for all odd-numbered exercises in the text. It is a great resource to help you understand how to solve those tough problems. Microsoft Excel Guide. This guide provides a list of exercises from the text that can be completed after each step-by-step Excel example. No prior knowledge of Excel is necessary. WebAssign. WebAssign, the most widely used homework system in higher education, offers instant feedback and repeatable problems—everything you could ask for in an online homework system. WebAssign’s homework system lets you practice and submit homework via the web. It is easy to use and loaded with extra resources. With this edition of Mathematical Applications for the Management, Life, and Social Sciences there are over 1,500 algorithmic homework exercises to use for practice and review. DVD Lecture Series. Comprehensive, instructional lecture presentations serve a number of uses. They are great if you need to catch up after missing a class, need to supplement online or hybrid instruction, or need material for self-study or review.

Resources for the Instructor WebAssign. Instant feedback, grading precision, and ease of use are just three reasons why WebAssign is the most widely used homework system in higher education. WebAssign’s homework delivery system lets instructors deliver, collect, grade, and record assignments via the web. With this edition of Mathematical Applications for the Management, Life, and Social Sciences there are over 1,500 algorithmic homework exercises to choose from. These algorithmic exercises are based on the section exercises from the textbook to ensure alignment with course goals. Instructor’s Complete Solutions Manual. The Instructor’s Complete Solutions Manual contains worked solutions for all exercises in the text. It also contains solutions to the special features in the text such as the Extended Applications and Group Projects. It is available at the Instructor’s Resource Center on the book’s website. Power Lecture. This comprehensive CD-ROM includes the Instructor’s Solutions Manual, PowerPoint slides, and the computerized test bank featuring algorithmically created questions to create, deliver, and customize tests. Computerized Test Bank. Create, deliver, and customize tests and study guides in minutes with this easy-to-use assessment software on CD. The thousands of algorithmic questions in the test bank are derived from the textbook exercises, ensuring consistency between exams and the book. JoinIn on TurningPoint. Enhance how your students interact with you, your lecture, and each other. Cengage Learning is now pleased to offer you book-speciﬁc content for Response Systems tailored to Mathematical Applications for the Management, Life, and Social Sciences, allowing you to transform your classroom and assess your students’ progress with instant in-class quizzes and polls.

xvi

● Preface

Acknowledgments We would like to thank the many people who helped us at various stages of revising this text. The encouragement, criticism, contributions, and suggestions that were offered were invaluable to us. Special thanks go to Victoria Sapko of Framingham State College for her contribution to the Inconvenient Truth Extended Application. We are also especially grateful to Dave Hipfel for his insights regarding bond pricing and his help with application exercises and Extended Applications. Once again we have been fortunate to have Helen Medley’s assistance with accuracy checking of the entire text and answer section during manuscript preparation and on the page proofs. As always, we continue to be impressed by and most appreciative of her work ethic, attention to detail, accuracy, and skill. For their reviews of draft manuscript and the many helpful comments that were offered, we would like to thank Jerry Allison Joy Beverly Elsie Campbell Vladimir Fomichov Mallie Hood Lance Lana Gabriel Mendoza Sam Obeid Elaine Russell Catherine Stanley Margaret Jeanne Trubek Dr. Henry L. Wyzinski

Black Hawk College University of Miami Angelo State University Southern New Hampshire University Paris Junior College University of Colorado DHSC El Paso Community College Richland College Angelina College Acadia University Emmanuel College Indiana University Northwest

Ronald J. Harshbarger James J. Reynolds

0 Algebraic Concepts This chapter provides a brief review of the algebraic concepts that will be used throughout the text. You may be familiar with these topics, but it may be helpful to spend some time reviewing them. In addition, each chapter after this one opens with a warm-up page that identifies prerequisite skills needed for that chapter. If algebraic skills are required, the warm-up cites their coverage in this chapter. Thus you will find that this chapter is a useful reference as you study later chapters. The topics and applications studied in this chapter include the following.

Sections

Applications

0.1 Sets Set operations Venn diagrams

Dow Jones Average, jobs growth

0.2 The Real Numbers Inequalities and intervals Absolute value

Income taxes, health statistics

0.3 Integral Exponents

Personal income, endangered species

0.4 Radicals and Rational Exponents Roots and fractional exponents Operations with radicals

Richter scale, half-life

0.5 Operations with Algebraic Expressions

Revenue, profit

0.6 Factoring Common factors Factoring trinomials

Simple interest, revenue

0.7 Algebraic Fractions Operations Complex fractions

Average cost, advertising and sales

2

● Chapter 0

Algebraic Concepts

0.1

Sets A set is a well-deﬁned collection of objects. We may talk about a set of books, a set of dishes, a set of students, or a set of individuals with a certain blood type. There are two ways to tell what a given set contains. One way is by listing the elements (or members) of the set (usually between braces). We may say that a set A contains 1, 2, 3, and 4 by writing A ⫽ 51, 2, 3, 46. To say that 4 is a member of set A, we write 4 僆 A. Similarly, we write 5 僆 A to denote that 5 is not a member of set A. If all the members of the set can be listed, the set is said to be a ﬁnite set. A ⫽ 51, 2, 3, 46 and B ⫽ 5x, y, z6 are examples of ﬁnite sets. When we do not wish to list all the elements of a ﬁnite set, we can use three dots to indicate the unlisted members of the set. For example, the set of even integers from 8 to 8952, inclusive, could be written as 58, 10, 12, 14, . . . , 89526

For an inﬁnite set, we cannot list all the elements, so we use the three dots. For example, N ⫽ 51, 2, 3, 4, . . .6 is an inﬁnite set. This set N is called the set of natural numbers. Another way to specify the elements of a given set is by description. For example, we may write D ⫽ 5x: x is a Ford automobile6 to describe the set of all Ford automobiles. Furthermore, F ⫽ 5y: y is an odd natural number6 is read “F is the set of all y such that y is an odd natural number.”

● EXAMPLE 1 Describing Sets Write the following sets in two ways. (a) The set A of natural numbers less than 6 (b) The set B of natural numbers greater than 10 (c) The set C containing only 3 Solution (a) A ⫽ 51, 2, 3, 4, 56 or A ⫽ 5x: x is a natural number less than 66 (b) B ⫽ 511, 12, 13, 14, . . .6 or B ⫽ 5x: x is a natural number greater than 106 (c) C ⫽ 536 or C ⫽ 5x: x ⫽ 36

Note that set C of Example 1 contains one member, 3; set A contains ﬁve members; and set B contains an inﬁnite number of members. It is possible for a set to contain no members. Such a set is called the empty set or the null set, and it is denoted by ⭋ or by { }. The set of living veterans of the War of 1812 is empty because there are no living veterans of that war. Thus 5x: x is a living veteran of the War of 18126 ⫽ ⭋

Special relations that may exist between two sets are deﬁned as follows. Relations Between Sets Definition

1. Sets X and Y are equal if they contain the same elements. 2. A is called a subset of B, which is written A 債 B, if every element of A is an element of B. The empty set is a subset of every set. Each set A is a subset of itself. 3. If C and D have no elements in common, they are called disjoint.

1. If X ⫽ 51, 2, 3, 46 and Y ⫽ 54, 3, 2, 16, then X ⫽ Y.

Example

2. If A ⫽ 51, 2, c, f 6 and B ⫽ 51, 2, 3, a, b, c, f 6, then A 債 B. Also, ⭋ 債 A, ⭋ 債 B, A 債 A, and B 債 B. 3. If C ⫽ 51, 2, a, b6 and D ⫽ 53, e, 5, c6, then C and D are disjoint.

3

0.1 Sets ●

In the discussion of particular sets, the assumption is always made that the sets under discussion are all subsets of some larger set, called the universal set U. The choice of the universal set depends on the problem under consideration. For example, in discussing the set of all students and the set of all female students, we may use the set of all humans as the universal set. We may use Venn diagrams to illustrate the relationships among sets. We use a rectangle to represent the universal set, and we use closed ﬁgures inside the rectangle to represent the sets under consideration. Figures 0.1–0.3 show such Venn diagrams.

U A

B

U

M

U

X

N

Y

Figure 0.1

Figure 0.2

Figure 0.3

B is a subset of A; B 債 A.

M and N are disjoint.

X and Y are not disjoint.

Set Operations

Set Intersection

The shaded portion of Figure 0.3 indicates where the two sets overlap. The set containing the members that are common to two sets is said to be the intersection of the two sets. A 傽 B ⫽ 5x: x 僆 A and x 僆 B6

The intersection of A and B, written A 傽 B, is deﬁned by

(a) If A ⫽ 52, 3, 4, 56 and B ⫽ 53, 5, 7, 9, 116, ﬁnd A 傽 B. (b) Which of A, B, and A 傽 B is a subset of A?

● EXAMPLE 2 Set Intersection

Solution (a) A 傽 B ⫽ 53, 56 because 3 and 5 are the common elements of A and B. Figure 0.4 shows the sets and their intersection. (b) A 傽 B and A are subsets of A.

U 9 2

A 4

B

3 5 7

Figure 0.4

●

Checkpoint

11

Let A ⫽ 52, 3, 5, 7, 116, B ⫽ 52, 4, 6, 8, 106, and C ⫽ 56, 10, 14, 18, 226. Use these sets to answer the following. 1. (a) Of which sets is 6 an element? (b) Of which sets is {6} an element? 2. Which of the following are true? (a) 2 僆 A (b) 2 僆 B (c) 2 僆 C (d) 5 僆 A (e) 5 僆 B 3. Which pair of A, B, and C is disjoint? 4. Which of ⭋, A, B, and C are subsets of (a) the set P of all prime numbers? (b) the set M of all multiples of 2? 5. Which of A, B, and C is equal to D ⫽ 5x 0 x ⫽ 4n ⫹ 2 for natural numbers 1 ⱕ n ⱕ 56?

4

● Chapter 0

Algebraic Concepts

The union of two sets is the set that contains all members of the two sets. Set Union

A ´ B ⫽ 5x: x 僆 A or x 僆 B (or both)6*

The union of A and B, written A ´ B, is deﬁned by

We can illustrate the intersection and union of two sets by the use of Venn diagrams. Figures 0.5 and 0.6 show Venn diagrams with universal set U represented by the rectangles and with sets A and B represented by the circles. The shaded region in Figure 0.5 represents A 傽 B, the intersection of A and B, and the shaded region in Figure 0.6—which consists of all parts of both circles—represents A ´ B.

U

A

U

A

B

Figure 0.5

B

Figure 0.6

If X ⫽ 5a, b, c, f 6 and Y ⫽ 5e, f, a, b6 , ﬁnd X ´ Y.

● EXAMPLE 3 Set Union Solution X ´ Y ⫽ 5a, b, c, e, f 6

Let A ⫽ 5x: x is a natural number less than 66 and B ⫽ 51, 3, 5, 7, 9, 116.

● EXAMPLE 4 Set Intersection and Union (a) Find A 傽 B.

(b) Find A ´ B.

Solution (a) A 傽 B ⫽ 51, 3, 56

(b) A ´ B ⫽ 51, 2, 3, 4, 5, 7, 9, 116

All elements of the universal set that are not contained in a set A form a set called the complement of A. Set Complement

A¿ ⫽ 5x: x 僆 U and x 僆 A6

The complement of A, written A¿, is deﬁned by

We can use a Venn diagram to illustrate the complement of a set. The shaded region of Figure 0.7 represents A¿, and the unshaded region of Figure 0.5 represents (A 傽 B)¿. *In mathematics, the word or means “one or the other or both.”

0.1 Sets ●

5

U

A A' Figure 0.7

If U ⫽ 5x 僆 N: x ⬍ 106, A ⫽ 51, 3, 66, and B ⫽ 51, 6, 8, 96, ﬁnd the following.

● EXAMPLE 5 Operations with Sets (a) A¿

(b) B¿

(c) (A 傽 B)¿

(d) A¿ ´ B¿

Solution (a) U ⫽ 51, 2, 3, 4, 5, 6, 7, 8, 96 so A¿ ⫽ 52, 4, 5, 7, 8, 96 (b) B¿ ⫽ 52, 3, 4, 5, 76 (c) A 傽 B ⫽ 51, 66 so (A 傽 B)¿ ⫽ 52, 3, 4, 5, 7, 8, 96 (d) A¿ ´ B¿ ⫽ 52, 4, 5, 7, 8, 96 ´ 52, 3, 4, 5, 76 ⫽ 52, 3, 4, 5, 7, 8, 96

●

Checkpoint

Given the sets U ⫽ 51, 2, 3, 4, 5, 6, 7, 8, 9, 106, A ⫽ 51, 3, 5, 7, 96, B ⫽ 52, 3, 5, 76, and C ⫽ 54, 5, 6, 7, 8, 9, 106 , ﬁnd the following. 6. A ´ B 7. B 傽 C 8. A¿

● EXAMPLE 6 Stocks Many local newspapers list “stocks of local interest.” Suppose that on a certain day, a prospective investor categorized 23 stocks according to whether ■

their closing price on the previous day was less than $50/share (set C)

■

their price-to-earnings ratio was less than 20 (set P)

■

their dividend per share was at least $1.50 (set D)

Of these 23 stocks, 16 12 8 3

belonged belonged belonged belonged

to to to to

set P set C set D both C and D

10 belonged to both C and P 7 belonged to both D and P 2 belonged to all three sets

Draw a Venn diagram that represents this information. Use the diagram to answer the following. (a) How many stocks ratios of less than (b) How many stocks (c) How many stocks

had closing prices of less than $50 per share or price-to-earnings 20? had none of the characteristics of set C, P, or D? had only dividends per share of at least $1.50?

Solution The Venn diagram for three sets has eight separate regions (see Figure 0.8(a) on the next page). To assign numbers from our data, we must begin with some information that refers to a single region, namely that two stocks belonged to all three sets (see Figure 0.8(b) on the next page). Because the region common to all three sets is also common to any pair,

6

● Chapter 0 Algebraic Concepts

we can next use the information about stocks that belonged to two of the sets (see Figure 0.8(c)). Finally, we can complete the Venn diagram (see Figure 0.8(d)). (a) We need to add the numbers in the separate regions that lie within C ´ P. That is, 18 stocks closed under $50 per share or had price-to-earnings ratios of less than 20. (b) There are 5 stocks outside the three sets C, D, and P. (c) Those stocks that had only dividends of at least $1.50 per share are inside D but outside both C and P. There are no such stocks.

C

P

U

P

C

U

C

U

2

D

D

(a)

(b)

C

P

2

P 1

8 5

U

5

1

1

8 2

1 5

0 D

D Figure 0.8

●

Checkpoint Solutions

(c)

(d)

1. (a) Sets B and C have 6 as an element. (b) None of A, B, or C has {6} as an element; {6} is itself a set, and the elements of A, B, and C are not sets. 2. (a) True (b) True (c) False; 2 僆 C (d) False; 5 僆 A (e) True 3. A and C are disjoint. 4. (a) ⭋ 債 P and A 債 P (b) ⭋ 債 M, B 債 M, C 債 M 5. C ⫽ D 6. A ´ B ⫽ 51, 2, 3, 5, 7, 96 7. B 傽 C ⫽ 55, 76 8. A¿ ⫽ 52, 4, 6, 8, 106

0.1 Exercises Use 僆 or 僆 to indicate whether the given object is an element of the given set in the following problems. 1. x 5x, y, z, a6 2. 3 51, 2, 3, 4, 5, 66 3. 12 51, 2, 3, 4, . . .6 4. 5 {x: x is a natural number greater than 5} 5. 6 {x: x is a natural number less than 6} 6. 3 ⭋

In Problems 7–10, write the following sets a second way. 7. {x: x is a natural number less than 8} 8. {x: x is a natural number greater than 6, less than 10} 9. 53, 4, 5, 6, 76 10. 57, 8, 9, 10, . . .6

In Problems 11 and 12, which of ⵰, A, and B are subsets of B? 11. A ⫽ 51, 2, 3, 46 and B ⫽ 51, 2, 3, 4, 5, 66

12. A ⫽ 5a, b, c, d6 and B ⫽ 5c, d, a, b6 13. Is A 債 B if A ⫽ 5a, b, c, d6 and B ⫽ 5a, b, d6? 14. Is A 債 B if A ⫽ 56, 8, 10, 126 and B ⫽ 56, 8, 10, 14, 186?

Use 債 notation to indicate which set is a subset of the other in the following problems. 15. C ⫽ 5a, b, 1, 2, 36, D ⫽ 5a, b, 16 16. E ⫽ 5x, y, a, b6, F ⫽ 5x, 1, a, y, b, 26 17. A ⫽ 56, 8, 7, 46, B ⫽ 58, 7, 6, 46 18. D ⫽ 5a, e, 1, 3, c6, F ⫽ 5e, a, c, 1, 36 In Problems 19–22, indicate whether the following pairs of sets are equal. 19. A ⫽ 5a, b, p, 136, B ⫽ 5a, p, 13, b6 20. A ⫽ 5x, g, a, b6, D ⫽ 5x, a, b, y6 21. D ⫽ 5x: x is a natural number less than 46, E ⫽ 51, 2, 3, 46 22. F ⫽ 5x: x is a natural number greater than 66, G ⫽ 57, 8, 9, . . .6 23. From the following list of sets, indicate which pairs of sets are disjoint. A ⫽ 51, 2, 3, 46 B ⫽ 5x: x is a natural number greater than 46 C ⫽ 54, 5, 6, . . .6 D ⫽ 51, 2, 36

24. If A and B are disjoint sets, what does A 傽 B equal? In Problems 25–28, ﬁnd A 傽 B, the intersection of sets A and B. 25. A ⫽ 52, 3, 4, 5, 66 and B ⫽ 54, 6, 8, 10, 126 26. A ⫽ 5a, b, c, d, e6 and B ⫽ 5a, d, e, f, g, h6 27. A ⫽ ⭋ and B ⫽ 5x, y, a, b6 28. A ⫽ 5x: x is a natural number less than 46 and B ⫽ 53, 4, 5, 66 In Problems 29–32, ﬁnd A ´ B, the union of sets A and B. 29. A ⫽ 51, 2, 4, 56 and B ⫽ 52, 3, 4, 56 30. A ⫽ 5a, e, i, o, u6 and B ⫽ 5a, b, c, d6 31. A ⫽ ⭋ and B ⫽ 51, 2, 3, 46 32. A ⫽ 5x: x is a natural number greater than 56 and B ⫽ 5x: x is a natural number less than 56 In Problems 33–44, assume that

A ⴝ {1, 3, 5, 8, 7, 2} B ⴝ {4, 3, 8, 10} C ⴝ {2, 4, 6, 8, 10} and that U is the universal set of natural numbers less than 11. Find the following. 33. A¿ 34. B¿ 35. A 傽 B¿ 36. A¿ 傽 B¿ 37. (A ´ B)¿ 38. (A 傽 B)¿

0.1 Sets ●

39. A¿ ´ B¿ 42. A 傽 (B¿ ´ C¿) 44. A 傽 (B ´ C)

40. (A¿ ´ B)¿

7

41. (A 傽 B¿) ´ C¿ 43. (A 傽 B¿)¿ 傽 C

The difference of two sets, A ⴚ B, is deﬁned as the set containing all elements of A except those in B. That is, A ⴚ B ⴝ A 傽 B¿. Find A ⴚ B for each pair of sets in Problems 45–48 if U ⴝ {1, 2, 3, 4, 5, 6, 7, 8, 9}. 45. A ⫽ 51, 3, 7, 96 and B ⫽ 53, 5, 8, 96 46. A ⫽ 51, 2, 3, 6, 96 and B ⫽ 51, 4, 5, 6, 76 47. A ⫽ 52, 1, 56 and B ⫽ 51, 2, 3, 4, 5, 66 48. A ⫽ 51, 2, 3, 4, 56 and B ⫽ 57, 8, 96 A P P L I C AT I O N S

49. Dow Jones Industrial Average The following table shows information about yearly lows, highs, and percentage changes for the years 2000 to 2006. Let L be the set of years where the low was greater than 8000. Let H be the set of years where the high was greater than 11,000. Let C be the years when the percentage change (from low to high) exceeded 35%. (a) List the elements of L, H, and C. (b) Is any of L, H, or C a subset of one of the others (besides itself)? (c) Write a verbal description of C¿. (d) Find H¿ ´ C¿ and describe it in words. (e) Find L¿ 傽 C and describe it in words.

Dow Jones Industrial Average Year

Low

High

% Change

2006 2005 2004 2003 2002 2001 2000

10,667.39 10,012.36 9749.99 7524.06 7286.27 8235.94 9796.03

12,510.57 10,940.50 10,854.54 10,453.92 10,635.65 11,332.92 11,722.98

17.3 9.3 11.3 38.9 45.9 37.6 19.7

Source: Dow Jones & Company, 2007

50. Job growth The number of jobs in 2000, the number projected in 2025, and the projected annual growth rate for jobs in some cities are shown in the table on the next page. Consider the following sets. A ⫽ set of cities with at least 2,000,000 jobs in 2000 or in 2025 B ⫽ set of cities with at least 1,500,000 jobs in 2000 C ⫽ set of cities with projected annual growth rate of at least 2.5%

8

● Chapter 0

Algebraic Concepts

(a) List A, B, and C (using the letters to represent the cities). (b) Is any of A, B, or C a subset of the other? (c) Find A 傽 C and describe the set in words. (d) Give a verbal description of B¿ .

Cities O (Orlando) M (Myrtle Beach) L (Atlanta) P (Phoenix) B (Boulder)

Jobs in 2000 (thousands)

Projected Jobs in 2025 (thousands)

Annual Rates of Increase (%)

1098 133 2715 1953 233

2207 256 4893 3675 420

2.83 2.64 2.38 2.56 2.38

Source: NPA Data Services, Inc.

National health care Suppose that the table below summarizes the opinions of various groups on the issue of national health care. Use this table for Problems 51 and 52. Whites

Nonwhites

Opinion

Rep.

Dem.

Rep.

Dem.

Total

Favor Oppose Total

100 250 350

250 150 400

30 10 40

200 10 210

580 420 1000

51. Identify the number of individuals in each of the following sets. (a) Republicans and those who favor national health care (b) Republicans or those who favor national health care (c) White Republicans or those who oppose national health care 52. Identify the number of individuals in each of the following sets. (a) Whites and those who oppose national health care (b) Whites or those who oppose national health care (c) Nonwhite Democrats and those who favor national health care 53. Languages A survey of 100 aides at the United Nations revealed that 65 could speak English, 60 could speak French, and 40 could speak both English and French. (a) Draw a Venn diagram representing the 100 aides. Use E to represent English-speaking aides and F to represent French-speaking aides. (b) How many aides are in E 傽 F? (c) How many aides are in E ´ F? (d) How many aides are in E 傽 F¿?

54. Advertising Suppose that a survey of 100 advertisers in U.S. News, These Times, and World found the following. 14 30 26 27 60 52 50

advertised advertised advertised advertised advertised advertised advertised

in in in in in in in

all three These Times and U.S. News World and U.S. News World and These Times These Times U.S. News World

Draw a Venn diagram representing this information and use it to answer the following. (a) How many advertised in none of these publications? (b) How many advertised only in These Times? (c) How many advertised in U.S. News or These Times? 55. College enrollments Records at a small college show the following about the enrollments of 100 ﬁrst-year students in mathematics, ﬁne arts, and economics. 38 take math 42 take ﬁne arts 20 take economics 4 take economics and ﬁne arts 15 take math and economics 9 take math and ﬁne arts 12 take math and economics but not ﬁne arts Draw a Venn diagram representing this information and label all the areas. Use this diagram to answer the following. (a) How many take none of these three courses? (b) How many take math or economics? (c) How many take exactly one of these three courses? 56. Survey analysis In a survey of the dining preferences of 110 dormitory students at the end of the spring semester, the following facts were discovered about Adam’s Lunch (AL), Pizza Tower (PT), and the Dining Hall (DH). 30 21 63 58 27 25 18

liked AL but not PT liked AL only liked AL liked PT liked DH liked PT and AL but not DH liked PT and DH

Draw a Venn diagram representing this survey and label all the areas. Use this diagram to answer the following. (a) How many liked PT or DH? (b) How many liked all three? (c) How many liked only DH?

0.2 The Real Numbers ●

57. Blood types Blood types are determined by the presence or absence of three antigens: A antigen, B antigen, and an antigen called the Rh factor. The resulting blood types are classiﬁed as follows: type type type type

A if the A antigen is present B if the B antigen is present AB if both the A and B antigens are present O if neither the A nor the B antigen is present

0.2

9

These types are further classiﬁed as Rh-positive if the Rh-factor antigen is present and Rh-negative otherwise. (a) Draw a Venn diagram that illustrates this classiﬁcation scheme. (b) Identify the blood type determined by each region of the Venn diagram (such as A⫹ to indicate type A, Rh-positive). (c) Use a library or another source to ﬁnd what percentage of the U.S. population has each blood type.

The Real Numbers This text uses the set of real numbers as the universal set. We can represent the real numbers along a line called the real number line. This number line is a picture, or graph, of the real numbers. Each point on the real number line corresponds to exactly one real number, and each real number can be located at exactly one point on the real number line. Thus, two real numbers are said to be equal whenever they are represented by the same point on the real number line. The equation a ⫽ b (a equals b) means that the symbols a and b represent the same real number. Thus, 3 ⫹ 4 ⫽ 7 means that 3 ⫹ 4 and 7 represent the same number. Table 0.1 lists special subsets of the real numbers.

TABLE 0.1 Subsets of the Set of Real Numbers Description Natural numbers Integers Rational numbers

Irrational numbers

Real numbers

Example (some elements shown)

{1, 2, 3, . . .} The counting numbers.

5. . . , ⫺2, ⫺1, 0, 1, 2, . . .6 The natural numbers, 0, and the negatives of the natural numbers. All numbers that can be written as the ratio of two integers, a兾b, with b ⫽ 0. These numbers have decimal representations that either terminate or repeat. Those real numbers that cannot be written as the ratio of two integers. Irrational numbers have decimal representations that neither terminate nor repeat. The set containing all rational and irrational numbers (the entire number line).

–3

–2

–1

0

1

2

3

4

5

–4

–3

–2

–1

0

1

2

3

4

–2

–1 −

–2

0

2

1 2

–1

0

2

–2

–1

7 4

1

− 3

–3

1

5 4

0

1

3 π

2

2

3

The properties of the real numbers are fundamental to the study of algebra. These properties follow.

Properties of the Real Numbers

Let a, b, and c denote real numbers. 1. Addition and multiplication are commutative. a⫹b⫽b⫹a

ab ⫽ ba

10

● Chapter 0

Algebraic Concepts

2. Addition and multiplication are associative. (a ⫹ b) ⫹ c ⫽ a ⫹ (b ⫹ c)

(ab)c ⫽ a(bc)

3. The additive identity is 0. a⫹0⫽0⫹a⫽a 4. The multiplicative identity is 1. aⴢ1⫽1ⴢa⫽a 5. Each element a has an additive inverse, denoted by ⫺a. a ⫹ (⫺a) ⫽ ⫺a ⫹ a ⫽ 0 Note that there is a difference between a negative number and the negative of a number. 6. Each nonzero element a has a multiplicative inverse, denoted by a⫺1. a ⴢ a⫺1 ⫽ a⫺1 ⴢ a ⫽ 1 Note that a⫺1 ⫽ 1兾a. 7. Multiplication is distributive over addition. a(b ⫹ c) ⫽ ab ⫹ ac Note that Property 5 provides the means to subtract by deﬁning a ⫺ b ⫽ a ⫹ (⫺b) and Property 6 provides a means to divide by deﬁning a ⫼ b ⫽ a ⴢ (1兾b). The number 0 has no multiplicative inverse, so division by 0 is undeﬁned.

Inequalities and Intervals

We say that a is less than b (written a ⬍ b) if the point representing a is to the left of the point representing b on the real number line. For example, 4 ⬍ 7 because 4 is to the left of 7 on the real number line. We may also say that 7 is greater than 4 (written 7 ⬎ 4). We may indicate that the number x is less than or equal to another number y by writing x ⱕ y. We may also indicate that p is greater than or equal to 4 by writing p ⱖ 4.

● EXAMPLE 1 Inequalities Use ⬍ or ⬎ notation to write (a) 6 is greater than 5. (c) 3 is to the left of 8 on the real number line.

(b) 10 is less than 15. (d) x is at most 12.

Solution (a) 6 ⬎ 5 (b) 10 ⬍ 15 (c) 3 ⬍ 8 (d) “x is at most 12” means it must be less than or equal to 12. Thus, x ⱕ 12. The subset of the real numbers consisting of all real numbers x that lie between a and b, excluding a and b, can be denoted by the double inequality a ⬍ x ⬍ b or by the open interval (a, b). It is called an open interval because neither of the endpoints is included in the interval. The closed interval [a, b] represents the set of all real numbers x satisfying a ⱕ x ⱕ b. Intervals containing one endpoint, such as (a, b] and [a, b), are called half-open intervals. We can use 3a, ⫹⬁) to represent the inequality x ⱖ a and (⫺⬁, a) to represent x ⬍ a. In each of these cases, the symbols ⫹⬁ and ⫺⬁ are not real numbers but represent the fact that x increases without bound (⫹⬁) or decreases without bound (⫺⬁). Table 0.2 summarizes three types of intervals.

11

0.2 The Real Numbers ● TABLE 0.2 Intervals Type of Interval Open interval

Half-open interval

Closed interval

●

Checkpoint

Absolute Value

Inequality Notation

Interval Notation

x⬎a

(a, ⬁ )

x⬍b

(⫺⬁ , b)

a⬍x⬍b

(a, b)

xⱖa

[a, ⬁ )

xⱕb

(⫺⬁ , b]

aⱕx⬍b

[a, b)

a⬍xⱕb

(a, b]

aⱕxⱕb

[a, b]

Graph a b a

b

a b a

b

a

b

a

b

1. Evaluate the following, if possible. For any that are meaningless, so state. 4 0 4 4⫺4 (a) (b) (c) (d) 0 4 4 4⫺4 2. For (a)–(d), write the inequality corresponding to the given interval and sketch its graph on a real number line. (a) (1, 3) (b) (0, 3] (c) 3⫺1, q) (d) (⫺q, 2) 3. Express each of the following inequalities in interval notation and name the type of interval. (a) 3 ⱕ x ⱕ 6 (b) ⫺6 ⱕ x ⬍ 4 Sometimes we are interested in the distance a number is from the origin (0) of the real number line, without regard to direction. The distance a number a is from 0 on the number line is the absolute value of a, denoted 0 a 0 . The absolute value of any nonzero number is positive, and the absolute value of 0 is 0.

● EXAMPLE 2 Absolute Value (a) 0 ⫺4 0

(b) 0 ⫹2 0

Evaluate the following. Solution (a) 0 ⫺4 0 ⫽ ⫹4 ⫽ 4 (c) 0 0 0 ⫽ 0

(c) 0 0 0

(d) 0 ⫺5 ⫺ 0 ⫺37

(b) 0 ⫹2 0 ⫽ ⫹2 ⫽ 2 (d) 0 ⫺5 ⫺ 0 ⫺3储 ⫽ 0 ⫺5 ⫺ 3 0 ⫽ 0 ⫺8 0 ⫽ 8

Note that if a is a nonnegative number, then 0 a 0 ⫽ a, but if a is negative, then 0 a 0 is the positive number (⫺a). Thus Absolute Value

0a0 ⫽ b

a ⫺a

if a ⱖ 0 if a ⬍ 0

In performing computations with real numbers, it is important to remember the rules for computations.

12

● Chapter 0

Algebraic Concepts

Operations with Real (Signed) Numbers Procedure

Example

1. (a) To add two real numbers with the same sign, add their absolute values and afﬁx their common sign. (b) To add two real numbers with unlike signs, ﬁnd the difference of their absolute values and afﬁx the sign of the number with the larger absolute value.

1. (a) (⫹5) ⫹ (⫹6) ⫽ ⫹11 1 2 3 1 a⫺ b ⫹ a⫺ b ⫽ ⫺ ⫽ ⫺ 6 6 6 2 (b) (⫺4) ⫹ (⫹3) ⫽ ⫺1 (⫹5) ⫹ (⫺3) ⫽ ⫹2 11 4 a⫺ b ⫹ (⫹1) ⫽ ⫺ 7 7

2. To subtract one real number from another, change the sign of the number being subtracted and proceed as in addition. 3. (a) The product of two real numbers with like signs is positive. (b) The product of two real numbers with unlike signs is negative. 4. (a) The quotient of two real numbers with like signs is positive. (b) The quotient of two real numbers with unlike signs is negative.

2. (⫺9) ⫺ (⫺8) ⫽ (⫺9) ⫹ (⫹8) ⫽ ⫺1 16 ⫺ (8) ⫽ 16 ⫹ (⫺8) ⫽ ⫹8 3. (a) (⫺3)(⫺4) ⫽ ⫹12 3 a⫹ b(⫹4) ⫽ ⫹3 4 (b) 5(⫺3) ⫽ ⫺15 (⫺3)(⫹4) ⫽ ⫺12 4. (a) (⫺14) ⫼ (⫺2) ⫽ ⫹7 ⫹36>4 ⫽ ⫹9 (b) (⫺28)>4 ⫽ ⫺7 45 ⫼ (⫺5) ⫽ ⫺9

When two or more operations with real numbers are indicated in an evaluation, it is important that everyone agree upon the order in which the operations are performed so that a unique result is guaranteed. The following order of operations is universally accepted.

Order of Operations 1. Perform operations within parentheses. 2. Find indicated powers (23 ⫽ 2 ⴢ 2 ⴢ 2 ⫽ 8). 3. Perform multiplications and divisions from left to right. 4. Perform additions and subtractions from left to right.

● EXAMPLE 3 Order of Operations Evaluate the following. (a) ⫺4 ⫹ 3

(b) ⫺42 ⫹ 3

(c) (⫺4 ⫹ 3)2 ⫹ 3

(d) 6 ⫼ 2(2 ⫹ 1)

Solution (a) ⫺1 (b) Note that with ⫺42 the power 2 is applied only to 4, not to ⫺4 (which would be written (⫺4)2). Thus ⫺42 ⫹ 3 ⫽ ⫺(42) ⫹ 3 ⫽ ⫺16 ⫹ 3 ⫽ ⫺13 (c) (⫺1)2 ⫹ 3 ⫽ 1 ⫹ 3 ⫽ 4 (d) 6 ⫼ 2(3) ⫽ (6 ⫼ 2)(3) ⫽ 3 ⴢ 3 ⫽ 9

13

0.2 The Real Numbers ● ●

Checkpoint

True or false: 4. ⫺(⫺5)2 ⫽ 25 5. 0 4 ⫺ 6 0 ⫽ 0 4 0 ⫺ 0 6 0 6. 9 ⫺ 2(2)(⫺10) ⫽ 7(2)(⫺10) ⫽ ⫺140 The text assumes that you have a scientiﬁc or graphing calculator. Discussions of some of the capabilities of graphing calculators and graphing utilities will be found throughout the text. Most scientiﬁc and graphing calculators use standard algebraic order when evaluating arithmetic expressions. Working outward from inner parentheses, calculations are performed from left to right. Powers and roots are evaluated ﬁrst, followed by multiplications and divisions and then additions and subtractions.

●

Checkpoint Solutions

1. (a) Meaningless. A denominator of zero means division by zero, which is undeﬁned. (b) 04 ⫽ 0. A numerator of zero (when the denominator is not zero) means the fraction has value 0. 4 (c) ⫽ 1 4 (d) Meaningless. The denominator is zero. 2. (a) 1 ⬍ x ⬍ 3 (b) 0 ⬍ x ⱕ 3 (c) ⫺1 ⱕ x ⬍ q

0

1

2

3

0

1

2

3

–2

–1

0

1

or x ⱖ ⫺1

3. (a) 33, 64; closed interval (b) 3⫺6, 4); half-open interval 4. False. ⫺(⫺5)2 ⫽ (⫺1)(⫺5)2 ⫽ (⫺1)(25) ⫽ ⫺25. Exponentiation has priority and applies only to ⫺5. 5. False. 0 4 ⫺ 6 0 ⫽ 0 ⫺2 0 ⫽ 2 and 0 4 0 ⫺ 0 6 0 ⫽ 4 ⫺ 6 ⫽ ⫺2. 6. False. Without parentheses, multiplication has priority over subtraction. 9 ⫺ 2(2)(⫺10) ⫽ 9 ⫺ 4(⫺10) ⫽ 9 ⫹ 40 ⫽ 49. (d) ⫺q ⬍ x ⬍ 2

or x ⬍ 2

–1

0

1

2

3

0.2 Exercises In Problems 1–2, indicate whether the given expression is one or more of the following types of numbers: rational, irrational, integer, natural. If the expression is meaningless, so state. ⫺p 9 4 1. (a) (b) ⫺9 (c) (d) 10 3 0 9 0 2. (a) (b) ⫺1.2916 (c) 1.414 (d) 6 6 Which part of 3. (a) 4. (a) 5. (a)

property of real numbers is illustrated in each Problems 3–6? 8⫹6⫽6⫹8 (b) 5(3 ⫹ 7) ⫽ 5(3) ⫹ 5(7) 6(4 ⴢ 5) ⫽ (6 ⴢ 4)(5) (b) ⫺15 ⫹ 0 ⫽ ⫺15 (b) 4 ⫹ (⫺4) ⫽ 0 ⫺e ⴢ 1 ⫽ ⫺e 3 3 2 3 6. (a) a b a b ⫽ 1 (b) (12)a b ⫽ a b (12) 2 3 4 4

Insert the proper sign ⬍, ⫽, or ⬎ to replace ⵧ in Problems 7–12. 7. ⫺14 ⵧ ⫺3 8. p ⵧ 3.14 1 1 1 5 9. 0.333 ⵧ 10. ⫹ ⵧ 3 3 2 6 11. ƒ ⫺3 ƒ ⫹ ƒ 5 ƒ ⵧ ƒ ⫺3 ⫹ 5 ƒ 12. ƒ ⫺9 ⫺ 3 ƒ ⵧ ƒ ⫺9 ƒ ⫹ ƒ 3 ƒ In Problems 13–24, evaluate each expression. 13. ⫺32 ⫹ 10 ⴢ 2 14. (⫺3)2 ⫹ 10 ⴢ 2 (4 ⫹ 2)2 4 ⫹ 22 15. 16. 2 2 16 ⫺ (⫺4) (⫺5)(⫺3) ⫺ (⫺2)(3) 17. 18. 8 ⫺ (⫺2) ⫺9 ⫹ 2 ƒ 3 ⫺ ƒ 4 ⫺ 11储 ƒ 5 ⫺ 2 ƒ ⫺ ƒ ⫺7 ƒ 19. 20. ƒ5 ⫺ 2ƒ ⫺ ƒ 52 ⫺ 32 ƒ

14

● Chapter 0

Algebraic Concepts

62 ⫺ 4(⫺3)(⫺2) (⫺3)2 ⫺ 2 ⴢ 3 ⫹ 6 22. 4 ⫺ 22 ⫹ 3 6 ⫺ 62 ⫼ 4 2 3 ⫺ 2(5 ⫺ 2) ⫺4 ⫹ 5 ⫺ 2 ⴢ 3 23. 24. 5 ⫺ 42 (⫺2)2 ⫺ 22 ⫹ 3 25. What part of the real number line corresponds to the interval (⫺⬁, ⬁)? 26. Write the interval corresponding to x ⱖ 0. 21.

In Problems 27–30, express each inequality or graph using interval notation, and name the type of interval. 27. 1 ⬍ x ⱕ 3 28. ⫺4 ⱕ x ⱕ 3 29. 0

2

4

6

8

10 12

30. –2

0

2

4

6

In Problems 31–34, write an inequality that describes each interval or graph. 31. 3⫺3, 5) 32. (⫺2, ⬁) 33. 0

2

4

6

8

0

2

4

5 6

8

34. In Problems 35–42, graph the subset of the real numbers that is represented by each of the following and write your answer in interval notation. 35. (⫺⬁, 4) 傽 (⫺3, ⬁) 36. 3⫺4, 17) 傽 3⫺20, 104 37. x ⬎ 4 and x ⱖ 0 38. x ⬍ 10 and x ⬍ ⫺1 39. 30, ⬁) ´ 3⫺1, 54 40. (⫺⬁, 4) ´ (0, 2) 41. x ⬎ 7 or x ⬍ 0 42. x ⬎ 4 and x ⬍ 0 In Problems 43–48, use your calculator to evaluate each of the following. List all the digits on your display in the answer. ⫺1 51.412 43. 44. 25916.8 127.01 45. (3.679)7 46. (1.28)10 (1.05)12 ⫺ 1 2500 47. 48. d 100 c (1.1)6 ⫺ 1 0.05 A P P L I C AT I O N S 49. Take-home pay A sales representative’s take-home pay is found by subtracting all taxes and retirement contributions from gross pay (which consists of salary plus commission). Given the following information, complete (a)–(c). Salary ⫽ $300.00 Commission ⫽ $788.91 Retirement ⫽ 5% of gross pay

Taxes: State ⫽ 5% of gross pay, Local ⫽ 1% of gross pay Federal withholding ⫽ 25% of (gross pay less retirement) Federal social security and Medicare ⫽ 7.65% of gross pay (a) Find the gross pay. (b) Find the amount of federal withholding. (c) Find the take-home pay. 50. Public health expenditures The expenditures E for government public health activities (in billions of dollars) can be approximated by E ⫽ 0.04t 2 ⫺ 0.56t ⫹ 1.29 where t is the number of years past 1960. (Source: Centers for Medicare and Medicaid Services) (a) What t-value represents the year 1995? (b) Actual expenditures for 1995 were $31.0 billion. What does the formula give as the 1995 approximation? (c) Predict the expenditures for 2012. 51. Health insurance coverage The percentage P of the U.S. population with no health insurance can be approximated quite accurately either by (1) P ⫽ 0.3179t ⫹ 13.85 or by (2) P ⫽ 0.0194t3 ⫺ 0.1952t2 ⫹ 0.8282t ⫹ 13.63 where t is the number of years past 2000. (Source: U.S. Census Bureau) (a) Both (1) and (2) closely approximate the data, but which is more accurate for 2006, when 15.8% of the population had no health insurance? (b) Use both formulas to predict the percentage of the U.S. population not covered in 2012. Which equation’s result seems more accurate? 52. Health statistics From data adapted from the National Center for Health Statistics, the height H in inches and age A in years for boys between 4 and 16 years of age are related according to H ⫽ 2.31A ⫹ 31.26 To account for normal variability among boys, normal height for a given age is ⫾5% of the height obtained from the equation. (a) Find the normal height range for a boy who is 10.5 years old, and write it as an inequality. (b) Find the normal height range for a boy who is 5.75 years old, and write it as an inequality.

0.3 Integral Exponents ●

53. Income taxes The 2007 tax brackets for a single person claiming one personal exemption are given in the following table. Taxable Income I

Tax Due T

$0–$7825 $7826–$31,850 $31,851–$77,100 $77,101–$160,850 $160,851–$349,700 Over $349,700

10% I $782.50 ⫹ 15%(I ⫺ 7825) $4386.25 ⫹ 25%(I ⫺ 31,850) $15,698.75 ⫹ 28%(I ⫺ 77,100) $39,148.75 ⫹ 33%(I ⫺ 160,850) $101,469.25 ⫹ 35%(I ⫺ 349,700)

15

(a) Write the last three taxable income ranges as inequalities. (b) If an individual has a taxable income of $31,850, calculate the tax due. Repeat this calculation for a taxable income of $77,100. (c) Write an interval that represents the amount of tax due for a taxable income between $31,850 and $77,100.

Source: Internal Revenue Service

0.3

Integral Exponents If $1000 is placed in a 5-year savings certiﬁcate that pays an interest rate of 10% per year, compounded annually, then the amount returned after 5 years is given by 1000(1.1)5 The 5 in this expression is an exponent. Exponents provide an easier way to denote certain multiplications. For example, (1.1)5 ⫽ (1.1)(1.1)(1.1)(1.1)(1.1) An understanding of the properties of exponents is fundamental to the algebra needed to study functions and solve equations. Furthermore, the deﬁnition of exponential and logarithmic functions and many of the techniques in calculus also require an understanding of the properties of exponents. For any real number a, a2 ⫽ a ⴢ a, a3 ⫽ a ⴢ a ⴢ a, and an ⫽ a ⴢ a ⴢ a ⴢ p ⴢ a

(n factors)

for any positive integer n. The positive integer n is called the exponent, the number a is called the base, and an is read “a to the nth power.” Note that 4an means 4(an), which is different from (4a) n. The 4 is the coefﬁcient of an in 4an. Note also that ⫺xn is not equivalent to (⫺x)n when n is even. For example, ⫺34 ⫽ ⫺81, but (⫺3)4 ⫽ 81. Some of the rules of exponents follow. Positive Integer Exponents

For any real numbers a and b and positive integers m and n, 1. am ⴢ an ⫽ am⫹n 2. For a ⫽ 0, 3. (ab)m ⫽ ambm a m am 4. a b ⫽ m b b 5. (am)n ⫽ amn

am⫺n if m ⬎ n am ⫽ c 1 if m ⫽ n an n⫺m 1兾a if m ⬍ n (b ⫽ 0)

16

● Chapter 0

Algebraic Concepts

● EXAMPLE 1 Positive Integer Exponents Use properties of positive integer exponents to rewrite each of the following. Assume all denominators are nonzero. (a)

56 54

(b)

x 4 (c) a b y

x2 x5

(d) (3x2y3)4

Solution 56 x2 1 1 (a) 4 ⫽ 56⫺4 ⫽ 52 (b) 5 ⫽ 5⫺2 ⫽ 3 5 x x x (d) (3x2y3)4 ⫽ 34(x2)4(y3)4 ⫽ 81x8y12

(e) 33 ⴢ 32 x 4 x4 (c) a b ⫽ 4 y y 3 (e) 3 ⴢ 32 ⫽ 33⫹2 ⫽ 35

For certain calculus operations, use of negative exponents is necessary in order to write problems in the proper form. We can extend the rules for positive integer exponents to all integers by deﬁning a0 and a⫺n. Clearly am ⴢ a0 should equal am⫹0 ⫽ am, and it will if a0 ⫽ 1. Zero Exponent

For any nonzero real number a, we deﬁne a0 ⫽ 1. We leave 00 undeﬁned. In Section 0.2, we deﬁned a⫺1 as 1兾a for a ⫽ 0, so we deﬁne a⫺n as (a⫺1)n. 1 n 1 a⫺n ⫽ (a⫺1)n ⫽ a b ⫽ n a a a ⫺n a ⫺1 n b n a b ⫽ ca b d ⫽ a b a b b

Negative Exponents

(a ⫽ 0) (a ⫽ 0, b ⫽ 0)

● EXAMPLE 2 Negative and Zero Exponents Write each of the following without exponents. (a) 6 ⴢ 30

(b) 6⫺2

Solution (a) 6 ⴢ 30 ⫽ 6 ⴢ 1 ⫽ 6

1 ⫺1 (c) a b 3

1 ⫺1 3 (c) a b ⫽ ⫽ 3 3 1 1 1 (e) (⫺4)⫺2 ⫽ ⫽ 2 (⫺4) 16

(b) 6⫺2 ⫽

2 ⫺4 (d) ⫺a b 3

(e) (⫺4)⫺2

1 1 ⫽ 2 6 36

2 ⫺4 3 4 ⫺81 (d) ⫺a b ⫽ ⫺a b ⫽ 3 2 16

As we’ll see in the chapter on the mathematics of ﬁnance (Chapter 6), negative exponents arise in ﬁnancial calculations when we have a future goal for an investment and want to know how much to invest now. For example, if money can be invested at 9%, compounded annually, then the amount we must invest now (which is called the present value) in order to have $10,000 in the account after 7 years is given by $10,000(1.09)⫺7. Calculations such as this are often done directly with a calculator. Using the deﬁnitions of zero and negative exponents enables us to extend the rules of exponents to all integers and to express them more simply.

0.3 Integral Exponents ●

Rules of Exponents

17

For real numbers a and b and integers m and n, 1. 3. 5. 7.

2. 4. 6. 8.

am ⴢ an ⫽ am⫹n (ab)m ⫽ ambm (a兾b)m ⫽ am兾bm (b ⫽ 0) a⫺n ⫽ 1兾an (a ⫽ 0)

am兾an ⫽ am⫺n (a ⫽ 0) (am)n ⫽ amn a0 ⫽ 1 (a ⫽ 0) (a兾b)⫺n ⫽ (b兾a)n (a, b ⫽ 0)

Throughout the remainder of the text, we will assume all expressions are deﬁned.

● EXAMPLE 3 Operations with Exponents Use the rules of exponents and the deﬁnitions of a0 and a⫺n to simplify each of the following with positive exponents. (a) 2(x2)⫺2

(b) x⫺2 ⴢ x⫺5

(c)

1 2 b⫽ 4 4 x x 1 ⫽ x⫺8⫺(⫺4) ⫽ x⫺4 ⫽ 4 x

x⫺8 x⫺4

Solution

(a) 2(x2)⫺2 ⫽ 2x⫺4 ⫽ 2a (c)

●

Checkpoint

x⫺8 x⫺4

(d) a

2x3 ⫺2 b 3x⫺5

(b) x⫺2 ⴢ x⫺5 ⫽ x⫺2⫺5 ⫽ x⫺7 ⫽ (d) a

1 x7

2x3 ⫺2 2x8 ⫺2 3 2 9 b ⫽ a b ⫽ a b ⫽ 16 3x⫺5 3 2x8 4x

1. Complete the following. (a) x3 ⴢ x8 ⫽ x?

(b) x ⴢ x4 ⴢ x⫺3 ⫽ x?

1 ⫽ x? x4 (f) (2x4y)3 ⫽ ?

(c)

(d) x24 ⫼ x⫺3 ⫽ x? (e) (x4)2 ⫽ x? 2. True or false: 1 ⫺1 (a) 3x⫺2 ⫽ 2 (b) ⫺x⫺4 ⫽ 4 (c) x⫺3 ⫽ ⫺x3 9x x 3. Evaluate the following, if possible. For any that are meaningless, so state. Assume x ⬎ 0. (a) 04 (b) 00 (c) x0 (d) 0x (e) 0⫺4 (f) ⫺5⫺2

● EXAMPLE 4 Rewriting a Quotient Write (x2y)兾(9wz3) with all factors in the numerator. Solution

x2y 1 1 1 1 ⫽ x2ya b ⫽ x2ya ba ba 3 b ⫽ x2y ⴢ 9⫺1w⫺1z⫺3 9wz3 9wz3 9 w z ⫺1 2 ⫺1 ⫺3 ⫽ 9 x yw z

● EXAMPLE 5 Rewriting with Positive Exponents Simplify the following so all exponents are positive. (a) (23x⫺4y5)⫺2

(b)

2x4(x2y)0 (4x⫺2y)2

18

● Chapter 0

Algebraic Concepts

Solution 1 1 x8 ⴢ x8 ⴢ 10 ⫽ 6 2 y 64y10 4 2 0 4 4 2x (x y) 2x ⴢ 1 2 x 1 2 x8 1 x8 (b) ⫽ ⫽ ⴢ ⴢ ⫽ ⴢ ⴢ ⫽ (4x⫺2y)2 42x⫺4y2 42 x⫺4 y2 16 1 y2 8y2 (a) (23x⫺4y5)⫺2 ⫽ 2⫺6x8y⫺10 ⫽

●

Checkpoint Solutions

1. (a) x3 ⴢ x8 ⫽ x3⫹8 ⫽ x11 (b) x ⴢ x4 ⴢ x⫺3 ⫽ x1⫹4⫹(⫺3) ⫽ x2 1 (c) 4 ⫽ x⫺4 (d) x24 ⫼ x⫺3 ⫽ x24⫺(⫺3) ⫽ x27 x (e) (x4)2 ⫽ x(4)(2) ⫽ x8 (f) (2x4y)3 ⫽ 23(x4)3y3 ⫽ 8x12y3 1 3 2. (a) False. 3x⫺2 ⫽ 3a 2 b ⫽ 2 x x 1 ⫺1 (b) True. ⫺x⫺4 ⫽ (⫺1)a 4 b ⫽ 4 x x 1 (c) False. x⫺3 ⫽ 3 x 3. (a) 04 ⫽ 0 (b) Meaningless. 00 is undeﬁned. 0 (c) x ⫽ 1 since x ⫽ 0 (d) 0x ⫽ 0 because x ⬎ 0 1 (e) Meaningless. 0⫺4 would be 4 , which is not deﬁned. 0 1 ⫺1 (f) ⫺5⫺2 ⫽ (⫺1)a 2 b ⫽ 5 25

0.3 Exercises Evaluate in Problems 1–8. Write all answers without using exponents. 1. (⫺4)4 2. ⫺53 3. ⫺26 4. (⫺2)5 2 3 2 3 5. 3⫺2 6. 6⫺1 7. ⫺a b 8. a b 2 3

In Problems 9–16, use rules of exponents to simplify the expressions. Express answers with positive exponents. 78 108 9. 65 ⴢ 63 10. 3 11. 7 109 54 12. ⫺2 3 13. (33)3 14. (2⫺3)⫺2 (5 ⴢ 5 ) 2 ⫺2 ⫺2 ⫺4 15. a b 16. a b 3 5

In Problems 17–20, simplify by expressing answers with positive exponents (x, y, z ⴝ 0). 17. (x2)⫺3 18. x⫺4 19. xy⫺2z0 20. (xy⫺2)0 In Problems 21–34, use the rules of exponents to simplify so that only positive exponents remain. 21. x3 ⴢ x4 22. a5 ⴢ a 23. x⫺5 ⴢ x3 8 x a5 24. y⫺5 ⴢ y⫺2 25. 4 26. ⫺1 x a y5 y⫺3 27. ⫺7 28. ⫺4 29. (x4)3 30. ( y3)⫺2 y y

31. (xy)2

32. (2m)3

33. a

2 4 b x5

34. a

8 3 b a3

In Problems 35–46, compute and simplify so that only positive exponents remain. 35. (2x⫺2y)⫺4 36. (⫺32x5)⫺3 ⫺3 2 5 ⫺4 37. (⫺8a b )(2a b ) 38. (⫺3m2y⫺1)(2m⫺3y⫺1) ⫺2 ⫺1 2 39. (2x ) ⫼ (x y ) 40. (⫺8a⫺3b2c) ⫼ (2a5b4) 3 ⫺3 x x⫺2 ⫺3 b 41. a ⫺2 b 42. a y y ⫺1 4x y⫺40 ⫺2 a⫺2b⫺1c⫺4 ⫺3 43. a 4 ⫺3 0 b 44. a ⫺2 4 ⫺10 b ab c 2 xy (2x)⫺2 2x⫺2 45. (a) (b) (2x)2 (2x)2 2x⫺2 2x⫺2 (c) (d) 2 2x (2x)⫺2 ⫺1 ⫺2 2 x 2⫺1x⫺2 46. (a) (b) (2x)2 2x2 (2x⫺2)⫺1 (2x⫺2)⫺1 (c) (d) (2x)⫺2 2x2 In many applications it is often necessary to write expressions in the form cxn, where c is a constant and n is an integer. In Problems 47–54, write the expressions in this form. 1 1 47. 48. 2 49. (2x)3 50. (3x)2 x x

0.3 Integral Exponents ●

51.

1 4x2

52.

53. a

3 2x4

⫺x 3 b 2

54. a

⫺x 2 b 3

In Problems 55–58, use a calculator to evaluate the indicated powers. 55. 1.24 56. (⫺3.7)3 57. (1.5)⫺5 58. (⫺0.8)⫺9 A P P L I C AT I O N S Compound interest If $P is invested for n years at rate i (as a decimal), compounded annually, the future value that accrues is given by S ⫽ P(1 ⫹ i)n, and the interest earned is I ⫽ S ⫺ P. In Problems 59–62, ﬁnd S and I for the given P, n, and i. 59. $1200 for 5 years at 12% 60. $1800 for 7 years at 10% 61. $5000 for 6 years at 11.5% 62. $800 for 20 years at 10.5% Present value If an investment has a goal (future value) of $S after n years, invested at interest rate i (as a decimal), compounded annually, then the present value P that must be invested is given by P ⫽ S(1 ⫹ i)⫺n. In Problems 63 and 64, ﬁnd P for the given S, n, and i. 63. $15,000 after 6 years at 11.5% 64. $80,000 after 20 years at 10.5% 65. Personal income For selected years from 1960 to 2005, billions of dollars of total U.S. personal income I can be approximated by the formula I ⫽ 456.1(1.074)t where t is the number of years past 1960. (Source: U.S. Department of Commerce) (a) What t-values correspond to the years 1970, 1990, and 2002? (b) The actual total personal incomes (in billions of dollars) for the years in part (a) were as follows. 1970

1990

2002

838.8

4878.6

8881.9

What does the formula predict for these years? (c) What does the formula predict for the total personal income in 2012? (d) Does this formula seem to indicate that total personal income doubles almost every 10 years? 66. Stock shares traded On the New York Stock Exchange (NYSE) for 1970–2006, the average daily shares traded S (in millions of shares) can be approximated by the formula S⫽

0.50274(1.1626)t

where t is the number of years past 1950. (Source: New York Stock Exchange)

19

(a) What t-values correspond to the years 1990, 2000, and 2006? (b) For the years in (a), the actual average millions of shares traded on the NYSE were as follows. 1990

2000

2006

156.777

1041.58

2343.16

What does the formula predict for these years? (c) Suppose in 2015 that a stock market average (such as the Dow Jones Industrial Average) dramatically soared or tumbled; do you think this formula’s predictions would be accurate, too low, or too high? Explain. 67. Endangered species The total number of endangered species y can be approximated by the formula y⫽

1883 1 ⫹ 7.892(1.097)⫺t

where t is the number of years past 1980. (Source: U.S. Fish and Wildlife Service) (a) The actual numbers of endangered species for selected years were as follows. 1990

2003

2007

442

987

1137

For each of these years, ﬁnd the number of endangered species predicted by the formula. Round your answer to the nearest integer. (b) How many more species does the formula estimate will be added to the endangered list for 2020 than the actual number given for 2007? (c) Why do you think the answer to (b) is smaller than the number of species added from 1990 to 2003? (d) Why is it reasonable for a formula such as this to have an upper limit that cannot be exceeded? Use large t-values in the formula to discover this formula’s upper limit. 68. Internet users The percent p of U.S. households with Internet service can be approximated by the equation p⫽

73.92 1 ⫹ 5.441(1.515)⫺t

where t is the number of years past 1995. (Source: U.S. Department of Commerce) (a) The percents of U.S. households with Internet service for selected years were as follows. 2001

2004

2007

50.0%

68.8%

70.2%

For each of these years, use the equation to ﬁnd the predicted percent of households with Internet service.

20

● Chapter 0

Algebraic Concepts

(that is, accurately approximated) by the formula

(b) In the three years from 2001 to 2004, the percent of households with Internet service increased by 18.8%. What percent increase does the equation predict from 2008 to 2011? Why do you think the 2008–2011 change is so different from the 2001–2004 change? (c) Why is it reasonable for a formula such as this to have an upper limit that cannot be exceeded? Use large t-values in the formula to discover this formula’s upper limit.

H ⫽ 30.58(1.102)t where t is the number of years past 1960. (Source: U.S. Department of Health and Human Services) (a) What t-value corresponds to 1970? (b) Approximate the national health care expenditure in 1970. (c) Approximate the national health care expenditure in 2005. (d) Estimate the national health care expenditure in 2015.

69. Health care expenditures The national health care expenditure H (in billions of dollars) can be modeled

0.4

Radicals and Rational Exponents

Roots

A process closely linked to that of raising numbers to powers is that of extracting roots. From geometry we know that if an edge of a cube has a length of x units, its volume is x3 cubic units. Reversing this process, we determine that if the volume of a cube is V cubic units, the length of an edge is the cube root of V, which is denoted 3 1 V units

3 When we seek the cube root of a number such as 8 (written 1 8), we are looking for a 3 real number whose cube equals 8. Because 23 ⫽ 8, we know that 1 8 ⫽ 2. Similarly, n 3 1 ⫺27 ⫽ ⫺3 because (⫺3)3 ⫽ ⫺27. The expression 1 a is called a radical, where 1 is the radical sign, n the index, and a the radicand. When no index is indicated, the index is assumed to be 2 and the expression is called a square root; thus 14 is the square root of 4 and represents the positive number whose square is 4. n Only one real number satisﬁes 1 a for a real number a and an odd number n; we call that number the principal nth root or, more simply, the nth root. For an even index n, there are two possible cases:

1. If a is negative, there is no real number equal to 1 a. For example, there are no real 4 numbers that equal 1⫺4 or 1 ⫺16 because there is no real number b such that n 2 4 b ⫽ ⫺4 or b ⫽ ⫺16. In this case, we say 1 a is not a real number. 2. If a is positive, there are two real numbers whose nth power equals a. For example, 32 ⫽ 9 and (⫺3)2 ⫽ 9. In order to have a unique nth root, we deﬁne the (principal) n nth root, 1 a, as the positive number b that satisﬁes bn ⫽ a. n

We summarize this discussion as follows. nth Root of a

1 a ⫽ b only if a ⫽ bn

The (principal) nth root of a real number is deﬁned as n

subject to the following conditions: a⫽0

n even n odd

n 1 a⫽0 n 1a ⫽ 0

a⬎0

n 1 a⬎0 n 1a ⬎ 0

a⬍0

n 1 a not real n 1a ⬍ 0

When we are asked for the root of a number, we give the principal root.

0.4 Radicals and Rational Exponents ●

21

● EXAMPLE 1 Roots 6 (a) 1 64

(b) ⫺ 116

3 (c) 1 ⫺8

(d) 1⫺16

Find the roots, if they are real numbers.

Solution 6 3 (a) 1 64 ⫽ 2 because 26 ⫽ 64 (b) ⫺ 116 ⫽ ⫺( 116) ⫽ ⫺4 (c) 1 ⫺8 ⫽ ⫺2 (d) 1⫺16 is not a real number because an even root of a negative number is not real.

Fractional Exponents

In order to perform evaluations on a calculator or to perform calculus operations, it is sometimes necessary to rewrite radicals in exponential form with fractional exponents. We have stated that for a ⱖ 0 and b ⱖ 0,

1a ⫽ b only if a ⫽ b2

This means that ( 1a)2 ⫽ b2 ⫽ a, or ( 1a)2 ⫽ a. In order to extend the properties of exponents to rational exponents, it is necessary to deﬁne a1兾2 ⫽ 1a

Exponent 1兾n

so that (a1兾2)2 ⫽ a

a1兾n ⫽ 1 a

For a positive integer n, we deﬁne

n

if

1 a exists n

Thus (a1兾n)n ⫽ a(1兾n)ⴢn ⫽ a. Because we wish the properties established for integer exponents to extend to rational exponents, we make the following deﬁnitions. Rational Exponents

For positive integer n and any integer m (with a ⫽ 0 when m ⱕ 0 and with m兾n in lowest terms): 1. am兾n ⫽ (a1兾n)m ⫽ ( 1 a)m n 2. am兾n ⫽ (am)1兾n ⫽ 1 am if a is nonnegative when n is even. n

Throughout the remaining discussion, we assume all expressions are real.

● EXAMPLE 2 Radical Form Write the following in radical form and simplify. (a) 163兾4

(b) y⫺3兾2

(c) (6m)2兾3

Solution 4 4 (a) 163兾4 ⫽ 1 163 ⫽ ( 1 16)3 ⫽ (2)3 ⫽ 8 1 1 (b) y⫺3兾2 ⫽ 3兾2 ⫽ y 1y3 3 3 (c) (6m)2兾3 ⫽ 1 (6m)2 ⫽ 1 36m2

22

● Chapter 0

Algebraic Concepts

● EXAMPLE 3 Fractional Exponents Write the following without radical signs. (a) 1x3

(b)

(a) 1x3 ⫽ x3兾2

3 (c) 1 (ab)3

1 b2 1

3

Solution

(b)

1

1

3

b2

⫽

1 b2兾3

⫽ b⫺2兾3

3 (c) 1 (ab)3 ⫽ (ab)3兾3 ⫽ ab

Our deﬁnition of am兾n guarantees that the rules for exponents will apply to fractional exponents. Thus we can perform operations with fractional exponents as we did with integer exponents.

● EXAMPLE 4 Operations with Fractional Exponents Simplify the following expressions. (a) a1兾2 ⴢ a1兾6

(b) a3兾4兾a1兾3

(c) (a3b)2兾3

(d) (a3兾2)1兾2

(e) a⫺1兾2 ⴢ a⫺3兾2

Solution (a) a1兾2 ⴢ a1兾6 ⫽ a1兾2⫹1兾6 ⫽ a3兾6⫹1兾6 ⫽ a4兾6 ⫽ a2兾3 (b) a3兾4兾a1兾3 ⫽ a3兾4⫺1兾3 ⫽ a9兾12⫺4兾12 ⫽ a5兾12 (c) (a3b)2兾3 ⫽ (a3)2兾3b2兾3 ⫽ a2b2兾3 (d) (a3兾2)1兾2 ⫽ a(3兾2)(1兾2) ⫽ a3兾4 (e) a⫺1兾2 ⴢ a⫺3兾2 ⫽ a⫺1兾2⫺3兾2 ⫽ a⫺2 ⫽ 1兾a2 ●

Checkpoint

Operations with Radicals

1. Which of the following are not real numbers? 3 4 5 8 (a) 1 ⫺64 (b) 1⫺64 (c) 10 (d) 1 1 (e) 1 ⫺1 (f) 1 ⫺1 1兾3 2兾5 ⫺3兾2 2. (a) Write as radicals: x , x , x 1 1 4 3 (b) Write with fractional exponents: 1 x ⫽ x?, ⫽ ? ⫽ x? 1x x 3. Evaluate the following. 15 (a) 82兾3 (b) (⫺8)2兾3 (c) 8⫺2兾3 (d) ⫺8⫺2兾3 (e) 1 71 4. Complete the following. (a) x ⴢ x1兾3 ⴢ x3 ⫽ x? (b) x2 ⫼ x1兾2 ⫽ x? (c) (x⫺2兾3)⫺3 ⫽ x? x4 3兾2 (d) x⫺3兾2 ⴢ x1兾2 ⫽ x? (e) x⫺3兾2 ⴢ x ⫽ x? (f) a 2 b ⫽ ? y 5. True or false: 8x2兾3 (a) ⫺1兾3 ⫽ 4x (b) (16x8y)3兾4 ⫽ 12x6y3兾4 x y3 1兾3 y x2 ⫺1兾3 (c) a 3 b ⫽ a 2 b ⫽ 2兾3 y x x We can perform operations with radicals by ﬁrst rewriting in exponential form, performing the operations with exponents, and then converting the answer back to radical form. Another option is to apply directly the following rules for operations with radicals.

0.4 Radicals and Rational Exponents ●

Given that 1 a and 1 b are real,* Rules for Radicals n

Examples

n

1. 1 an ⫽ ( 1 a)n ⫽ a n n n 2. 1 a ⴢ 1 b ⫽ 1 ab n 1a a (b ⫽ 0) 3. n ⫽ n 1b Bb n

23

5 5 5 1. 1 6 ⫽ (1 6)5 ⫽ 6 3 3 3 3 3 2. 1 2 1 4 ⫽ 1 8⫽ 1 2 ⫽2 118 18 3. ⫽ ⫽ 19 ⫽ 3 B2 12

n

*Note that this means a ⱖ 0 and b ⱖ 0 if n is even.

1 a is not real, and Rule 1 does not apply. For example, 1⫺2 is not a real number, and n

Let us consider Rule 1 for radicals more carefully. Note that if n is even and a ⬍ 0, then

1(⫺2)2 ⫽ ⫺2 because 1(⫺2)2 ⫽ 14 ⫽ 2 ⫽ ⫺(⫺2)

We can generalize this observation as follows: If a ⬍ 0, then 1a2 ⫽ ⫺a ⬎ 0, so

1a2 ⫽ b

a ⫺a

if a ⱖ 0 if a ⬍ 0

1a2 ⫽ 冟a冟

This means

● EXAMPLE 5 Simplifying Radicals 3 3 (a) 1 8

5 5 (b) 1 x

Simplify:

(c) 1x2

7 (d) 3 1 (3x2 ⫹ 4)3 4 7

Solution 3 3 (a) 1 8 ⫽ 8 by Rule 1 for radicals 5 5 7 (b) 1 x ⫽x (c) 1x2 ⫽ 冟x冟 (d) 3 1 (3x2 ⫹ 4)3 47 ⫽ (3x2 ⫹ 4)3

Up to now, to simplify a radical has meant to ﬁnd the indicated root. More generally, n a radical expression 1 x is considered simpliﬁed if x has no nth powers as factors. Rule 2 n n n for radicals ( 1 a ⴢ 1 b ⫽ 1 ab) provides a procedure for simplifying radicals.

● EXAMPLE 6 Simplifying Radicals (a) 148x5y6

3 (b) 1 72a3b4

Simplify the following radicals; assume the expressions are real numbers. (y ⱖ 0)

Solution (a) To simplify 148x5y6, we ﬁrst factor 48x5y6 into perfect-square factors and other factors. Then we apply Rule 2.

148x5y6 ⫽ 116 ⴢ 3 ⴢ x4xy6 ⫽ 116 1x4 1y6 13x ⫽ 4x2y3 13x

3 3 3 3 3 3 3 3 3 1 72a3b4 ⫽ 1 8 ⴢ 9a3b3b ⫽ 1 8ⴢ 1 a ⴢ1 b ⴢ1 9b ⫽ 2ab 1 9b

(b) In this case, we factor 72a3b4 into factors that are perfect cubes and other factors.

Rule 2 for radicals also provides a procedure for multiplying two roots with the same index.

24

● Chapter 0

Algebraic Concepts

● EXAMPLE 7 Multiplying Radicals 3 3 (a) 1 2xy ⴢ 1 4x2y

(b) 18xy3z 14x2y3z2

Multiply the following and simplify the answers, assuming nonnegative variables. Solution 3 3 3 3 3 3 3 3 2 3 2 (a) 1 2xy ⴢ 1 4x2y ⫽ 1 2xy ⴢ 4x2y ⫽ 1 8x3y2 ⫽ 1 8ⴢ 1 x ⴢ1 y ⫽ 2x 1 y (b) 18xy3z 14x2y3z2 ⫽ 132x3y6z3 ⫽ 116x2y6z2 12xz ⫽ 4xy3z 12xz Rule 3 for radicals ( 1 a兾 1 b ⫽ 1 a兾b) indicates how to ﬁnd the quotient of two roots with the same index. n

n

n

● EXAMPLE 8 Dividing Radicals 3 1 32 3 14

116a3x 12ax

Find the quotients and simplify the answers, assuming nonnegative variables. (a)

(b)

Solution 3 1 32 32 3 (a) 3 ⫽ 3 ⫽1 8⫽2 B 4 14

Rationalizing

(b)

116a3x 16a3x ⫽ ⫽ 18a2 ⫽ 2a 12 B 2ax 12ax

Occasionally, we wish to express a fraction containing radicals in an equivalent form that contains no radicals in the denominator. This is accomplished by multiplying the numerator and the denominator by the expression that will remove the radical. This process is called rationalizing the denominator.

● EXAMPLE 9 Rationalizing Denominators Express each of the following with no radicals in the denominator. (Rationalize each denominator.) (a)

1x 15

(b)

118xy 2x

(x, y ⬎ 0)

(c)

1 2x2 3x

3

(x ⫽ 0)

Solution (a) We wish to create a perfect square under the radical in the denominator.

1x 15

ⴢ

1x 15 1x ⫽ x 1x

12xy 12xy 2x 12xy 2x 12xy ⫽ ⫽ ⫽ 2 2 6xy 3y 12xy 136x y

118xy (c) We wish to create a perfect cube under the radical in the denominator. (b)

2x

ⴢ

3 1 2x2

3x

●

Checkpoint

6. Simplify: 7 7 (a) 1 x

ⴢ

3 3 3 3 1 4x 3x 1 4x 3x 1 4x 31 4x ⫽ ⫽ ⫽ 3 3 2x 2 1 4x 1 8x3

5 (b) 3 1 (x2 ⫹ 1)2 4 5

7. Rationalize the denominator of

(c) 112xy2 ⴢ 13x2y

15x x

if x ⫽ 0.

0.4 Radicals and Rational Exponents ●

25

It is also sometimes useful, especially in calculus, to rationalize the numerator of a fraction. For example, we can rationalize the numerator of 3 1 4x2

3 by multiplying the numerator and denominator by 1 2x, which creates a perfect cube under the radical:

3x

3 1 4x2

3x ●

Checkpoint Solutions

ⴢ

3 3 1 2x 1 8x3 2x 2 ⫽ ⫽ ⫽ 3 3 3 3 1 2x 3x 1 2x 3x 1 2x 3 1 2x

8 1. Only even roots of negatives are not real numbers. Thus 1⫺64 and 1 ⫺1 are not real numbers. 1 1 3 5 2. (a) x1兾3 ⫽ 1 x, x2兾5 ⫽ 1 x2, x⫺3兾2 ⫽ 3兾2 ⫽ x 1x3 1 1 4 3 ⫽ ⫽ x⫺1兾2 (b) 1 x ⫽ x3兾4, 1x x1兾2 3 3 3. (a) 82兾3 ⫽ ( 1 8)2 ⫽ 22 ⫽ 4 (b) (⫺8)2兾3 ⫽ ( 1 ⫺8)2 ⫽ (⫺2)2 ⫽ 4 1 1 1 1 (c) 8⫺2兾3 ⫽ 2兾3 ⫽ (d) ⫺8⫺2兾3 ⫽ ⫺a 2兾3 b ⫽ ⫺ 8 4 8 4 15 (e) 1 71 ⫽ (71)1兾15 ⬇ 1.32867 4. (a) x ⴢ x1兾3 ⴢ x3 ⫽ x1⫹1兾3⫹3 ⫽ x13兾3 (b) x2 ⫼ x1兾2 ⫽ x2⫺1兾2 ⫽ x3兾2 (c) (x⫺2兾3)⫺3 ⫽ x(⫺2兾3)(⫺3) ⫽ x2 (d) x⫺3兾2 ⴢ x1兾2 ⫽ x⫺3兾2⫹1兾2 ⫽ x⫺1 x4 3兾2 (x4)3兾2 x6 (e) x⫺3兾2 ⴢ x ⫽ x⫺3兾2⫹1 ⫽ x⫺1兾2 (f) a 2 b ⫽ 2 3兾2 ⫽ 3 y (y ) y 8x2兾3 ⫽ 8x2兾3 ⴢ x1兾3 ⫽ 8x2兾3⫹1兾3 ⫽ 8x 5. (a) False. x⫺1兾3 4 (16x8y)3兾4 ⫽ 163兾4(x8)3兾4y3兾4 ⫽ ( 1 16)3x6y3兾4 ⫽ 8x6y3兾4 (b) False. (c) True. 7 6. (a) 1 x7 ⫽ x 5 (b) 3 1 (x2 ⫹ 1)2 4 5 ⫽ (x2 ⫹ 1)2 (c) 112xy2 ⴢ 13x2y ⫽ 136x3y3 ⫽ 136x2y2 ⴢ xy ⫽ 136x2y2 1xy ⫽ 6xy 1xy x x 15x x 15x 15x 7. ⫽ ⴢ ⫽ ⫽ 5x 5 15x 15x 15x

0.4 Exercises Unless stated otherwise, assume all variables are nonnegative and all denominators are nonzero. In Problems 1–8, ﬁnd the powers and roots, if they are real numbers. 1. (a) 1256兾9 (b) 11.44 5 4 2. (a) 1 ⫺323 (b) 1 ⫺165 3. (a) 163兾4 (b) (⫺16)⫺3兾2 4. (a) ⫺27⫺1兾3 (b) 323兾5 8 ⫺2兾3 4 3兾2 5. a b 6. a b 27 9 7. (a) 82兾3 (b) (⫺8)⫺2兾3 8. (a) 8⫺2兾3 (b) ⫺82兾3

In Problems 9 and 10, rewrite each radical with a fractional exponent, and then approximate the value with a calculator. 12 9 9. 1 (6.12)4 10. 1 4.96 In Problems 11–14, replace each radical with a fractional exponent. Do not simplify. 3 5 4 5 3 11. 1m3 12. 1 x 13. 1 m2n5 14. 1 x In Problems 15–18, write in radical form. Do not simplify. 15. x7兾6 16. y11兾5 17. ⫺(1兾4)x⫺5兾4 18. ⫺x⫺5兾3

26

● Chapter 0

Algebraic Concepts

In Problems 19–32, use the properties of exponents to simplify each expression so that only positive exponents remain. 19. y1兾4 ⴢ y1兾2 20. x2兾3 ⴢ x1兾5 21. z3兾4 ⴢ z4 22. x⫺2兾3 ⴢ x2 23. y⫺3兾2 ⴢ y⫺1 24. z⫺2 ⴢ z5兾3 y⫺5兾2 x1兾3 x⫺1兾2 25. ⫺2兾3 26. ⫺3兾2 27. ⫺2兾5 x x y x4兾9 28. 1兾12 29. (x2兾3)3兾4 30. (x4兾5)3 x 31. (x⫺1兾2)2 32. (x⫺2兾3)⫺2兾5 In Problems 33–38, simplify each expression by using the properties of radicals. Assume nonnegative variables. 3 33. 164x4 34. 1 35. 1128x4y5 ⫺64x6y3 3 3 36. 1 54x5x8 37. 1 40x8y5 38. 132x5y In Problems 39–46, perform the indicated operations and simplify. 3 3 39. 112x3y ⴢ 13x2y 40. 1 16x2y ⴢ 1 3x2y 5 3 2 10 41. 163x y ⴢ 128x y 42. 110xz ⴢ 130x17z 112x3y12 1250xy7z4 43. 44. 127xy2 118x17y2 3 4 9 5 1 ⫺16x3y4 1 32a b 45. 4 46. 3 1 162a17 1 128y2 In Problems 47–50, use properties of exponents and radicals to determine a value for x that makes each statement true. 47. (A9)x ⫽ A 48. (B20)x ⫽ B 7 49. ( 1 R)x ⫽ R 50. ( 1T 3)x ⫽ T In Problems 51–56, rationalize each denominator and then simplify. 1m2x 51. 12兾3 52. 15兾8 53. 1mx2 3 4 3 2 5x w 1m x 1 mx3 54. 55. 3 56. 4 14xw2 1 mx5 1 y2z5 In calculus it is frequently important to write an expression in the form cxn, where c is a constant and n is a rational number. In Problems 57–60, write each expression in this form. ⫺2 ⫺2 3 57. 3 58. 4 59. 3x 1x 60. 1x ⴢ 1 x 2 31x 3 1 x3 In calculus problems, the answers are frequently expected to be in a simple form with a radical instead of an exponent. In Problems 61–64, write each expression with radicals. ⫺1 ⫺3兾2 3 4 1 61. x1兾2 62. x1兾3 63. x⫺1兾2 64. x 2 3 2 2 A P P L I C AT I O N S 65. Richter scale The Richter scale reading for an earthquake measures its intensity (as a multiple of some min-

imum intensity used for comparison). The intensity I corresponding to a Richter scale reading R is given by I ⫽ 10R (a) A quake measuring 8.5 on the Richter scale would be severe. Express the intensity of such a quake in exponential form and in radical form. (b) Find the intensity of a quake measuring 8.5. (c) The San Francisco quake that occurred during the 1989 World Series measured 7.1, and the 1906 San Francisco quake (which devastated the city) measured 8.25. Calculate the ratio of these intensities (larger to smaller). 66. Sound intensity The intensity of sound I (as a multiple of the average minimum threshold of hearing intensity) is related to the decibel level D (or loudness of sound) according to I ⫽ 10D兾10 (a) Express 10D兾10 using radical notation. (b) The background noise level of a relatively quiet room has a decibel reading of 32. Find the intensity I1 of this noise level. (c) A decibel reading of 140 is at the threshold of pain. If I2 is the intensity of this threshold and I1 is the intensity found in (b), express the ratio I2兾I1 as a power of 10. Then approximate this ratio. 67. Investment If $1000 is invested at r% compounded annually, the future value S of the account after two and a half years is given by S ⫽ 1000a1 ⫹

r 2.5 r 5兾2 b ⫽ 1000a1 ⫹ b 100 100

(a) Express this equation with radical notation. (b) Find the value of this account if the interest rate is 6.6% compounded annually. 68. Life span Life expectancy in the United States can be approximated with the equation L ⫽ 29x0.21 where x is the number of years that the birth year is past 1900. (Source: National Center for Health Statistics) (a) Express this equation with radical notation. (b) Use the equation to estimate the life expectancy for a person born in 2015. 69. Population The population P of India (in billions) for 2000–2050 can be approximated by the equation P ⫽ 0.924t 0.13 where t ⬎ 0 is the number of years past 2000. (Source: United Nations)

0.5 Operations with Algebraic Expressions ●

(a) Express this equation with radical notation. (b) Does this equation predict a greater increase from 2005 to 2010 or from 2045 to 2050? What might explain this difference? 70. Transportation The percent p of paved roads and streets in the United States can be approximated with the equation p ⫽ 6.75t 0.55 where t is the number of years past 1940. (Source: U.S. Department of Transportation) (a) Express this equation with radical notation. (b) Does this equation estimate a greater percent change during the decade of the 1970s or during the decade from 2000 to 2010? What might explain this? (c) When can you be certain this equation is no longer valid? Half-life In Problems 71 and 72, use the fact that the quantity of a radioactive substance after t years is given by q ⴝ q0(2 ⴚt兾k), where q0 is the original amount of radioactive material and k is its half-life (the number of years it takes for half the radioactive substance to decay). 71. The half-life of strontium-90 is 25 years. Find the amount of strontium-90 remaining after 10 years if q0 ⫽ 98 kg. 72. The half-life of carbon-14 is 5600 years. Find the amount of carbon-14 remaining after 10,000 years if q0 ⫽ 40.0 g.

0.5

27

73. Population growth Suppose the formula for the growth of the population of a city for the next 10 years is given by P ⫽ P0(2.5)ht where P0 is the population of the city at the present time and P is the population t years from now. If h ⫽ 0.03 and P0 ⫽ 30,000, ﬁnd P when t ⫽ 10. 74. Advertising and sales Suppose it has been determined that the sales at Ewing Gallery decline after the end of an advertising campaign, with daily sales given by S ⫽ 2000(2⫺0.1x) where S is in dollars and x is the number of days after the campaign ends. What are the daily sales 10 days after the end of the campaign? 75. Company growth The growth of a company can be described by the equation t

N ⫽ 500(0.02)0.7

where t is the number of years the company has been in existence and N is the number of employees. (a) What is the number of employees when t ⫽ 0? (This is the number of employees the company has when it starts.) (b) What is the number of employees when t ⫽ 5?

Operations with Algebraic Expressions In algebra we are usually dealing with combinations of real numbers (such as 3, 6兾7, and ⫺ 12) and letters (such as x, a, and m). Unless otherwise speciﬁed, the letters are symbols used to represent real numbers and are sometimes called variables. An expression obtained by performing additions, subtractions, multiplications, divisions, or extractions of roots with one or more real numbers or variables is called an algebraic expression. Unless otherwise speciﬁed, the variables represent all real numbers for which the algebraic expression is a real number. Examples of algebraic expressions include 3x ⫹ 2y,

x3y ⫹ y , x⫺1

and

1x ⫺ 3

Note that the variable x cannot be negative in the expression 1x ⫺ 3 and that (x3y ⫹ y)兾(x ⫺ 1) is not a real number when x ⫽ 1, because division by 0 is undeﬁned. Any product of a real number (called the coefﬁcient) and one or more variables to powers is called a term. The sum of a ﬁnite number of terms with nonnegative integer powers on the variables is called a polynomial. If a polynomial contains only one variable x, then it is called a polynomial in x.

28

● Chapter 0

Algebraic Concepts

Polynomial in x

The general form of a polynomial in x is anxn ⫹ an⫺1xn⫺1 ⫹ p ⫹ a1x ⫹ a0 where each coefﬁcient ai is a real number for i ⫽ 0, 1, 2, . . . , n. If an ⫽ 0, the degree of the polynomial is n, and an is called the leading coefﬁcient. The term a0 is called the constant term. Thus 4x3 ⫺ 2x ⫺ 3 is a third-degree polynomial in x with leading coefﬁcient 4 and constant term ⫺3. If two or more variables are in a term, the degree of the term is the sum of the exponents of the variables. The degree of a nonzero constant term is zero. Thus the degree of 4x2y is 2 ⫹ 1 ⫽ 3, the degree of 6xy is 1 ⫹ 1 ⫽ 2, and the degree of 3 is 0. The degree of a polynomial containing one or more variables is the degree of the term in the polynomial having the highest degree. Therefore, 2xy ⫺ 4x ⫹ 6 is a second-degree polynomial. A polynomial containing two terms is called a binomial, and a polynomial containing three terms is called a trinomial. A single-term polynomial is a monomial.

Operations with Algebraic Expressions

Because monomials and polynomials represent real numbers, the properties of real numbers can be used to add, subtract, multiply, divide, and simplify polynomials. For example, we can use the Distributive Law to add 3x and 2x. 3x ⫹ 2x ⫽ (3 ⫹ 2)x ⫽ 5x Similarly, 9xy ⫺ 3xy ⫽ (9 ⫺ 3)xy ⫽ 6xy. Terms with exactly the same variable factors are called like terms. We can add or subtract like terms by adding or subtracting the coefﬁcients of the variables.

● EXAMPLE 1 Adding Polynomials Compute (4xy ⫹ 3x) ⫹ (5xy ⫺ 2x). Solution (4xy ⫹ 3x) ⫹ (5xy ⫺ 2x) ⫽ 4xy ⫹ 3x ⫹ 5xy ⫺ 2x ⫽ 9xy ⫹ x Subtraction of polynomials uses the Distributive Law to remove the parentheses.

● EXAMPLE 2 Subtracting Polynomials Compute (3x2 ⫹ 4xy ⫹ 5y2 ⫹ 1) ⫺ (6x2 ⫺ 2xy ⫹ 4). Solution Removing the parentheses yields 3x2 ⫹ 4xy ⫹ 5y2 ⫹ 1 ⫺ 6x2 ⫹ 2xy ⫺ 4 which simpliﬁes to ⫺3x2 ⫹ 6xy ⫹ 5y2 ⫺ 3 Using the rules of exponents and the Commutative and Associative Laws for multiplication, we can multiply and divide monomials, as the following example shows.

0.5 Operations with Algebraic Expressions ●

29

● EXAMPLE 3 Products and Quotients Perform the indicated operations. (a) (8xy3)(2x3y)(⫺3xy2)

(b) ⫺15x2y3 ⫼ (3xy5)

Solution (a) 8 ⴢ 2 ⴢ (⫺3) ⴢ x ⴢ x3 ⴢ x ⴢ y3 ⴢ y ⴢ y2 ⫽ ⫺48x5y6 ⫺15x2y3 15 x2 y3 1 5x (b) ⫽⫺ ⴢ ⫽ ⫺5 ⴢ x ⴢ 2 ⫽ ⫺ 2 ⴢ 5 3xy 3 x y5 y y Symbols of grouping are used in algebra in the same way as they are used in the arithmetic of real numbers. We have removed parentheses in the process of adding and subtracting polynomials. Other symbols of grouping, such as brackets, [ ], are treated the same as parentheses. When there are two or more symbols of grouping involved, we begin with the innermost and work outward. Simplify 3x2 ⫺ 32x ⫺ (3x2 ⫺ 2x)4.

● EXAMPLE 4 Symbols of Grouping

Solution

3x2 ⫺ 32x ⫺ (3x2 ⫺ 2x)4 ⫽ 3x2 ⫽ 3x2 ⫽ 3x2 ⫽ 6x2

⫺ 32x ⫺ 3x2 ⫹ 2x4 ⫺ 34x ⫺ 3x2 4 ⫺ 4x ⫹ 3x2 ⫺ 4x

By the use of the Distributive Law, we can multiply a binomial by a monomial. For example, x(2x ⫹ 3) ⫽ x ⴢ 2x ⫹ x ⴢ 3 ⫽ 2x2 ⫹ 3x We can extend the Distributive Law to multiply polynomials with more than two terms. For example, 5(x ⫹ y ⫹ 2) ⫽ 5x ⫹ 5y ⫹ 10

● EXAMPLE 5 Distributive Law Find the following products. (a) ⫺4ab(3a2b ⫹ 4ab2 ⫺ 1)

(b) (4a ⫹ 5b ⫹ c)ac

Solution (a) ⫺4ab(3a2b ⫹ 4ab2 ⫺ 1) ⫽ ⫺12a3b2 ⫺ 16a2b3 ⫹ 4ab (b) (4a ⫹ 5b ⫹ c)ac ⫽ 4a ⴢ ac ⫹ 5b ⴢ ac ⫹ c ⴢ ac ⫽ 4a2c ⫹ 5abc ⫹ ac2 The Distributive Law can be used to show us how to multiply two polynomials. Consider the indicated multiplication (a ⫹ b)(c ⫹ d). If we ﬁrst treat the sum (a ⫹ b) as a single quantity, then two successive applications of the Distributive Law gives (a ⫹ b)(c ⫹ d) ⫽ (a ⫹ b) ⴢ c ⫹ (a ⫹ b) ⴢ d ⫽ ac ⫹ bc ⫹ ad ⫹ bd Thus we see that the product can be found by multiplying (a ⫹ b) by c, multiplying (a ⫹ b) by d, and then adding the products. This is frequently set up as follows.

30

● Chapter 0

Algebraic Concepts

Product of Two Polynomials Procedure

Example

To multiply two polynomials:

Multiply (3x ⫹ 4xy ⫹ 3y) by (x ⫺ 2y).

1. Write one of the polynomials above the other.

1. 3x ⫹ 4xy ⫹ 3y x ⫺ 2y

2. Multiply each term of the top polynomial by each term of the bottom one, and write the similar terms of the product under one another.

2. 3x2 ⫹ 4x2y ⫹ 3xy ⫺ 6xy ⫺ 8xy2 ⫺ 6y2

3. Add like terms to simplify the product.

3. 3x2 ⫹ 4x2y ⫺ 3xy ⫺ 8xy2 ⫺ 6y2

● EXAMPLE 6 The Product of Two Polynomials Multiply (4x2 ⫹ 3xy ⫹ 4x) by (2x ⫺ 3y). Solution 4x2 ⫹ 3xy ⫹ 4x 2x ⫺ 3y 8x3 ⫹ 6x2y ⫹ 8x2 ⫺12x2y ⫺ 9xy2 ⫺ 12xy 8x3 ⫺ 6x2y ⫹ 8x2 ⫺ 9xy2 ⫺ 12xy Because the multiplications we must perform often involve binomials, the following special products are worth remembering. Special Products

A. (x ⫹ a)(x ⫹ b) ⫽ x2 ⫹ (a ⫹ b)x ⫹ ab B. (ax ⫹ b)(cx ⫹ d) ⫽ acx2 ⫹ (ad ⫹ bc)x ⫹ bd It is easier to remember these two special products if we note their structure. We can obtain these products by ﬁnding the products of the First terms, Outside terms, Inside terms, and Last terms, and then adding the results. This is called the FOIL method of multiplying two binomials.

● EXAMPLE 7 Products of Binomials Multiply the following. (a) (x ⫺ 4)(x ⫹ 3)

(b) (3x ⫹ 2)(2x ⫹ 5)

Solution First

Outside

Inside

Last

(a) (x ⫺ 4)(x ⫹ 3) ⫽ (x2) ⫹ (3x) ⫹ (⫺4x) ⫹ (⫺12) ⫽ x2 ⫺ x ⫺ 12 (b) (3x ⫹ 2)(2x ⫹ 5) ⫽ (6x2) ⫹ (15x) ⫹ (4x) ⫹ (10) ⫽ 6x2 ⫹ 19x ⫹ 10 Additional special products are as follows:

0.5 Operations with Algebraic Expressions ●

Additional Special Products

C. D. E. F. G.

(x ⫹ a)2 ⫽ x2 ⫹ 2ax ⫹ a2 (x ⫺ a)2 ⫽ x2 ⫺ 2ax ⫹ a2 (x ⫹ a)(x ⫺ a) ⫽ x2 ⫺ a2 (x ⫹ a)3 ⫽ x3 ⫹ 3ax2 ⫹ 3a2x ⫹ a3 (x ⫺ a)3 ⫽ x3 ⫺ 3ax2 ⫹ 3a2x ⫺ a3

31

binomial squared binomial squared difference of two squares binomial cubed binomial cubed

● EXAMPLE 8 Special Products Multiply the following. (a) (b) (c) (d) (e)

(x ⫹ 5)2 (3x ⫺ 4y)2 (x ⫺ 2)(x ⫹ 2) (x2 ⫺ y3)2 (x ⫹ 4)3

Solution (a) (x ⫹ 5)2 ⫽ x2 ⫹ 2(5)x ⫹ 25 ⫽ x2 ⫹ 10x ⫹ 25 (b) (3x ⫺ 4y)2 ⫽ (3x)2 ⫺ 2(3x)(4y) ⫹ (4y)2 ⫽ 9x2 ⫺ 24xy ⫹ 16y2 (c) (x ⫺ 2)(x ⫹ 2) ⫽ x2 ⫺ 4 (d) (x2 ⫺ y3)2 ⫽ (x2)2 ⫺ 2(x2)(y3) ⫹ (y3)2 ⫽ x4 ⫺ 2x2y3 ⫹ y6 (e) (x ⫹ 4)3 ⫽ x3 ⫹ 3(4)(x2) ⫹ 3(42)(x) ⫹ 43 ⫽ x3 ⫹ 12x2 ⫹ 48x ⫹ 64 ●

Checkpoint

1. Remove parentheses and combine like terms: 9x ⫺ 5x(x ⫹ 2) ⫹ 4x2 2. Find the following products. (a) (2x ⫹ 1)(4x2 ⫺ 2x ⫹ 1) (b) (x ⫹ 3)2 (c) (3x ⫹ 2)(x ⫺ 5) (d) (1 ⫺ 4x)(1 ⫹ 4x) All algebraic expressions can represent real numbers, so the techniques used to perform operations on polynomials and to simplify polynomials also apply to other algebraic expressions.

● EXAMPLE 9 Operations with Algebraic Expressions 3 13 ⫹ 4x 1y ⫺ 5 13 ⫺ 11x 1y ⫺ ( 13 ⫺ x 1y) x3兾2(x1兾2 ⫺ x⫺1兾2) (x1兾2 ⫺ x1兾3)2 ( 1x ⫹ 2)( 1x ⫺ 2)

Perform the indicated operations. (a) (b) (c) (d)

Solution (a) We remove parentheses and then combine the terms containing 13 and the terms containing x 1y. (3 ⫺ 5 ⫺ 1) 13 ⫹ (4 ⫺ 11 ⫹ 1)x 1y ⫽ ⫺3 13 ⫺ 6x 1y

x3兾2(x1兾2

(b) ⫺ x⫺1兾2) ⫽ x3兾2 ⴢ x1兾2 ⫺ x3兾2 ⴢ x⫺1兾2 ⫽ x2 ⫺ x 1兾2 1兾3 (c) (x ⫺ x )2 ⫽ (x1兾2)2 ⫺ 2x1兾2x1兾3 ⫹ (x1兾3)2 ⫽ x ⫺ 2x5兾6 ⫹ x2兾3 (d) ( 1x ⫹ 2)( 1x ⫺ 2) ⫽ ( 1x)2 ⫺ (2)2 ⫽ x ⫺ 4

32

● Chapter 0

Algebraic Concepts

In later chapters we will need to write problems in a simpliﬁed form so that we can perform certain operations on them. We can often use division of one polynomial by another to obtain the simpliﬁcation, as shown in the following procedure.

Division of Polynomials Procedure

Example

To divide one polynomial by another:

Divide 4x3 ⫹ 4x2 ⫹ 5 by 2x2 ⫹ 1.

1. Write both polynomials in descending powers of a variable. Include missing terms with coefﬁcient 0 in the dividend.

1. 2x2 ⫹ 1冄4x3 ⫹ 4x2 ⫹ 0x ⫹ 5

2. (a) Divide the highest power term of the divisor into the highest power term of the dividend, and write this partial quotient above the dividend. Multiply the partial quotient times the divisor, write the product under the dividend, and subtract, getting a new dividend. (b) Repeat until the degree of the new dividend is less than the degree of the divisor. Any remainder is written over the divisor and added to the quotient.

2x 2. (a) 2x ⫹ 1冄4x3 ⫹ 4x2 ⫹ 0x ⫹ 5 4x3 ⫹ 2x 2 4x ⫺ 2x ⫹ 5 2

2x ⫹ 2 (b) 2x2 ⫹ 1冄4x3 ⫹ 4x2 ⫹ 0x ⫹ 5 4x3 ⫹ 2x 4x2 ⫺ 2x ⫹ 5 4x2 ⫹2 ⫺ 2x ⫹ 3 Degree (⫺2x ⫹ 3) ⬍ degree (2x2 ⫹ 1) ⫺2x ⫹ 3 Quotient: 2x ⫹ 2 ⫹ 2x2 ⫹ 1

● EXAMPLE 10 Division of Polynomials Divide (4x3 ⫺ 13x ⫺ 22) by (x ⫺ 3),

x ⫽ 3.

Solution 4x2 ⫹ 12x ⫹ 23 x ⫺ 3冄4x3 ⫹ 0x2 ⫺ 13x ⫺ 22 4x3 ⫺ 12x2 12x2 ⫺ 13x ⫺ 22 12x2 ⫺ 36x 23x ⫺ 22 23x ⫺ 69 47

0x2 is inserted so that each power of x is present.

The quotient is 4x2 ⫹ 12x ⫹ 23, with remainder 47, or 4x2 ⫹ 12x ⫹ 23 ⫹

●

Checkpoint

47 x⫺3

3. Use long division to ﬁnd (x3 ⫹ 2x ⫹ 7) ⫼ (x ⫺ 4).

0.5 Operations with Algebraic Expressions ●

Spreadsheet Note

33

One important use of algebraic expressions is to describe relationships among quantities. For example, the expression “one more than a number” could be written as n ⫹ 1, where n represents an arbitrary number. This ability to represent quantities or their interrelationships algebraically is one of the keys to using spreadsheets. For more details regarding spreadsheets, see the Excel Guide for Finite Mathematics and Applied Calculus by Revathi Narasimhan that accompanies this text. Each cell in a spreadsheet has an address based on its row and column (see Table 0.3). These cell addresses can act like variables in an algebraic expression, and the “ﬁll down” or “ﬁll across” capabilities update this cell referencing while maintaining algebraic relationships. For example, we noted previously that if $1000 is invested in an account that earns 10%, compounded annually, then the future value of the account after n years is given by $1000(1.1)n. We can track the future value by starting with the spreadsheet shown in Table 0.4. TABLE 0.3 A 1 2 3 4 5

B

cell A1 cell A2 cell A3 . .

C

cell B1 cell B2 cell B3

D .

cell C1 cell C2 cell C3

TABLE 0.4 A 1 2 3

Year 1

B Future value ⫽ 1000*(1.1) ^ (A2)

gives

⎯⎯→

⫽A2⫹1

A 1 2 3

B Future value

Year 1 2

1100

The use of the ⫽ sign to begin a cell entry indicates an algebraic expression whose variables are other cells. In cell A3 of Table 0.4, typing the entry ⫽A2⫹1 creates a referencing scheme based on the algebraic expression n ⫹ 1, but with cell A2 acting as the variable. From this beginning, highlighting cell A3 and using the “ﬁll down” command updates the referencing from cell to cell and creates a counter for the number of years (see Table 0.5). TABLE 0.5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

A Year

B Future value 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1100 1210 1331 1464.1 1610.51 1771.561 1948.7171 2143.58881 2357.947691 2593.7424601 2853.1167061 3138.4283767 3452.2712144 3797.4983358 4177.2481694

C Interest earned 100 210 331 464.1 610.51 771.561 948.7171 1143.58881 1357.947691 1593.74246 1853.116706 2138.428377 2452.271214 2797.498336 3177.248169

34

● Chapter 0

Algebraic Concepts

The future value is given by a formula, so we use that formula in the cell but replace the variable with the cell name where a value of that variable can be found. The “ﬁll down” command updates the referencing and hence the values used. The ﬁrst 15 years of the spreadsheet begun in Table 0.4 are shown in Table 0.5, along with a new column for interest earned. Because the interest earned is found by subtracting the original investment of $1000 from the future value, cell C2 contains the entry ⫽B2⫺1000, and those below it are obtained by using “ﬁll down.” The entries for future value and interest earned also could be expressed in dollars and cents by rounding appropriately. ■ ●

Checkpoint Solutions

1. 9x ⫺ 5x(x ⫹ 2) ⫹ 4x2 ⫽ 9x ⫺ 5x2 ⫺ 10x ⫹ 4x2 ⫽ ⫺x2 ⫺ x Note that without parentheses around 9x ⫺ 5x, multiplication has priority over subtraction. 2. (a) 4x2 ⫺ 2x ⫹ 1 2x ⫹ 1 8x3 ⫺ 4x2 ⫹ 2x 4x2 ⫺ 2x ⫹ 1 3 8x ⫹1 (b) (x ⫹ 3)2 ⫽ x2 ⫹ 2(3x) ⫹ 32 ⫽ x2 ⫹ 6x ⫹ 9 (c) (3x ⫹ 2)(x ⫺ 5) ⫽ 3x2 ⫺ 15x ⫹ 2x ⫺ 10 ⫽ 3x2 ⫺ 13x ⫺ 10 Note that this is different from 16x2 ⫺ 1. (d) (1 ⫺ 4x)(1 ⫹ 4x) ⫽ 1 ⫺ 16x2 x2 ⫹ 4x ⫹ 18 79 3. x ⫺ 4冄x3 ⫹ 0x2 ⫹ 2x ⫹ 7 The answer is x2 ⫹ 4x ⫹ 18 ⫹ . x⫺4 3 2 x ⫺ 4x 4x2 ⫹ 2x ⫹ 7 4x2 ⫺ 16x 18x ⫹ 7 18x ⫺ 72 79

0.5 Exercises For each polynomial in Problems 1–4, (a) give the degree of the polynomial, (b) give the coefﬁcient (numerical) of the highest-degree term, (c) give the constant term, and (d) decide whether it is a polynomial of one or several variables. 1. 10 ⫺ 3x ⫺ x2 2. 5x4 ⫺ 2x9 ⫹ 7 3. 7x2y ⫺ 14xy3z 4. 2x5 ⫹ 7x2y3 ⫺ 5y6 The expressions in Problems 5 and 6 are polynomials with the form anxn ⴙ anⴚ1xnⴚ1 ⴙ . . . ⴙ a1x ⴙ a0, where n is a positive integer. Complete the following. 5. For 2x5 ⫺ 3x2 ⫺ 5, (a) 2 ⫽ a? (b) a3 ⫽ ? (c) ⫺3 ⫽ a? (d) a0 ⫽ ? 6. For 5x3 ⫺ 4x ⫺ 17, (a) a3 ⫽ ? (b) a1 ⫽ ? (c) a2 ⫽ ? (d) ⫺17 ⫽ a?

In Problems 7–10, evaluate each algebraic expression at the indicated values of the variables. 7. 4x ⫺ x2 at x ⫽ ⫺2 8. 3x2 ⫺ 4y2 ⫺ 2xy at x ⫽ 3 and y ⫽ ⫺4 2x ⫺ y 9. 2 at x ⫽ ⫺5 and y ⫽ ⫺3 x ⫺ 2y 16y 10. at y ⫽ ⫺3 1⫺y 11. Evaluate 1.98T ⫺ 1.09(1 ⫺ H)(T ⫺ 58) ⫺ 56.8 when T ⫽ 74.7 and H ⫽ 0.80. 0.083i 12. Evaluate R c d when R ⫽ 100,000, 1 ⫺ (1 ⫹ 0.083i)⫺n i ⫽ 0.07, n ⫽ 360. In Problems 13–20, simplify by combining like terms. 13. (16pq ⫺ 7p2) ⫹ (5pq ⫹ 5p2) 14. (3x3 ⫹ 4x2y2) ⫹ (3x2y2 ⫺ 7x3)

0.5 Operations with Algebraic Expressions ●

15. 16. 17. 18. 19. 20.

(4m2 ⫺ 3n2 ⫹ 5) ⫺ (3m2 ⫹ 4n2 ⫹ 8) (4rs ⫺ 2r2s ⫺ 11rs2) ⫺ (11rs2 ⫺ 2rs ⫹ 4r2s) ⫺38 ⫺ 4(q ⫹ 5) ⫹ q4 x3 ⫹ 33x ⫺ (x3 ⫺ 3x)4 x2 ⫺ 3x ⫺ (x2 ⫺ 1) ⫹ 1 ⫺ (1 ⫺ x2)4 ⫹ x y3 ⫺ 3y2 ⫺ (y3 ⫹ y2)4 ⫺ 3y3 ⫹ (1 ⫺ y2)4

In Problems 21–58, perform the indicated operations and simplify. 21. (5x3)(7x2) 22. (⫺3x2y)(2xy3)(4x2y2) 3 2 2 23. (39r s ) ⫼ (13r s) 24. (⫺15m3n) ⫼ (5mn4) 2 2 25. ax (2x ⫹ ax ⫹ ab) 26. ⫺3(3 ⫺ x2) 27. (3y ⫹ 4)(2y ⫺ 3) 28. (4x ⫺ 1)(x ⫺ 3) 29. 6(1 ⫺ 2x2)(2 ⫺ x2) 30. 2(x3 ⫹ 3)(2x3 ⫺ 5) 2 31. (4x ⫹ 3) 32. (2y ⫹ 5)2 2 1 33. ax2 ⫺ b 34. (x3y3 ⫺ 0.3)2 2 35. 9(2x ⫹ 1)(2x ⫺ 1) 36. 3(5y ⫹ 2)(5y ⫺ 2) 2 2 37. (0.1 ⫺ 4x)(0.1 ⫹ 4x) 38. a ⫹ xb a ⫺ xb 3 3 39. (0.1x ⫺ 2)(x ⫹ 0.05) 40. (6.2x ⫹ 4.1)(6.2x ⫺ 4.1) 41. (x ⫺ 2)(x2 ⫹ 2x ⫹ 4) 42. (a ⫹ b)(a2 ⫺ ab ⫹ b2) 43. (x3 ⫹ 5x)(x5 ⫺ 2x3 ⫹ 5) 44. (x3 ⫺ 1)(x7 ⫺ 2x4 ⫺ 5x2 ⫹ 5) 45. (a) (3x ⫺ 2)2 ⫺ 3x ⫺ 2(3x ⫺ 2) ⫹ 5 (b) (3x ⫺ 2)2 ⫺ (3x ⫺ 2)(3x ⫺ 2) ⫹ 5 46. (a) (2x ⫺ 3)(3x ⫹ 2) ⫺ (5x ⫺ 2)(x ⫺ 3) (b) 2x ⫺ 3(3x ⫹ 2) ⫺ 5x ⫺ 2(x ⫺ 3) 47. (18m2n ⫹ 6m3n ⫹ 12m4n2) ⫼ (6m2n) 48. (16x2 ⫹ 4xy2 ⫹ 8x) ⫼ (4xy) 49. (24x8y4 ⫹ 15x5y ⫺ 6x7y) ⫼ (9x5y2) 50. (27x2y2 ⫺ 18xy ⫹ 9xy2) ⫼ (6xy) 51. (x ⫹ 1)3 52. (x ⫺ 3)3 3 53. (2x ⫺ 3) 54. (3x ⫹ 4)3 3 55. (x ⫹ x ⫺ 1) ⫼ (x ⫹ 2) 56. (x5 ⫹ 5x ⫺ 7) ⫼ (x ⫹ 1) 57. (x4 ⫹ 3x3 ⫺ x ⫹ 1) ⫼ (x2 ⫹ 1) 58. (x3 ⫹ 5x2 ⫺ 6) ⫼ (x2 ⫺ 2) ˛

In Problems 59–66, perform the indicated operations with expressions involving fractional exponents and radicals, and then simplify. 59. x1兾2(x1兾2 ⫹ 2x3兾2) 60. x⫺2兾3(x5兾3 ⫺ x⫺1兾3) 61. (x1兾2 ⫹ 1)(x1兾2 ⫺ 2) 62. (x1兾3 ⫺ x1兾2)(4x2兾3 ⫺ 3x3兾2) 63. ( 1x ⫹ 3)( 1x ⫺ 3) 64. (x1兾5 ⫹ x1兾2)(x1兾5 ⫺ x1兾2) 65. (2x ⫹ 1)1兾2 3(2x ⫹ 1)3兾2 ⫺ (2x ⫹ 1)⫺1兾2 4 66. (4x ⫺ 3)⫺5兾3 3(4x ⫺ 3)8兾3 ⫹ 3(4x ⫺ 3)5兾3 4

35

A P P L I C AT I O N S 67. Revenue A company sells its product for $55 per unit. Write an expression for the amount of money received (revenue) from the sale of x units of the product. 68. Proﬁt Suppose a company’s revenue R (in dollars) from the sale of x units of its product is given by R ⫽ 215x Suppose further that the total costs C (in dollars) of producing those x units is given by C ⫽ 65x ⫹ 15,000 (a) If proﬁt is revenue minus cost, ﬁnd an expression for the proﬁt from the production and sale of x units. (b) Find the proﬁt received if 1000 units are sold. 69. Rental A rental truck costs $49.95 for a day plus 49¢ per mile. (a) If x is the number of miles driven, write an expression for the total cost of renting the truck for a day. (b) Find the total cost of the rental if it was driven 132 miles. 70. Cell phones Cell Pro makes cell phones and has total weekly costs of $1500 for rent, utilities, and equipment plus labor and material costs of $18.50 for each phone it makes. (a) If x represents the number of phones produced and sold, write an expression for Cell Pro’s weekly costs. (b) If Cell Pro sells the phones to dealers for $45.50 each, write an expression for the weekly total revenue for the phones. (c) Cell Pro’s weekly proﬁt is the total revenue minus the total cost. Write an expression for Cell Pro’s weekly proﬁt. 71. Investments Suppose that you have $4000 to invest, and you invest x dollars at 10% and the remainder at 8%. Write expressions in x that represent (a) the amount invested at 8%, (b) the interest earned on the x dollars at 10%, (c) the interest earned on the money invested at 8%, (d) the total interest earned. 72. Medications Suppose that a nurse needs 10 cc (cubic centimeters) of a 15.5% solution (that is, a solution that is 15.5% ingredient) of a certain medication, which must be obtained by mixing x cc of a 20% solution and y cc of a 5% solution. Write expressions involving x for (a) y, the amount of 5% solution, (b) the amount of ingredient in the x cc of 20% solution, (c) the amount of ingredient in the 5% solution, (d) the total amount of ingredient in the mixture.

36

● Chapter 0

Algebraic Concepts

73. Package design The volume of a rectangular box is given by V ⫽ (length)(width)(height). If a rectangular piece of cardboard that is 10 in. by 15 in. has a square with sides of length x cut from each corner (see Figure 0.9), and if the sides are folded up along the dotted lines to form a box, what expression of x would represent the volume?

75. Area Suppose we were interested in different areas for rectangles for which the sum of the length and width is 50. If the length and width must be whole numbers, the beginning of a spreadsheet listing the possibilities might look like this: A 1 2 3 4

15 x x

B

length

C area = (l)(w)

width 49 48 47

1 2 3

49 96 141

10

Figure 0.9

Problems 74–77 use spreadsheets. 74. Coin mix Suppose an individual has 125 coins, with some nickels and some dimes. We can use a spreadsheet to investigate the values for various numbers of nickels and dimes. Suppose we set up a spreadsheet with column A for the number of nickels, column B for the number of dimes, and column C for the value of the coins. Then the beginning of the spreadsheet might look like this: A 1 2 3 4

B

nickels

C

dimes 0 1 2

value 125 124 123

12.5 12.45 12.4

(a) With 0 as the entry in cell A2, what cell A3 entry references A2 and could be used to ﬁll column A with the possible numbers of nickels? (b) Knowing that the total number of coins is 125, how do we ﬁnd the number of dimes if we know the number of nickels? (c) Use the result in (b) to write an entry for cell B2 that is referenced to A2 and that could be used to ﬁll column B with the correct number of dimes for each possible number of nickels in column A. (d) How do you use the entries in columns A and B to ﬁnd the corresponding value for column C? Write a rule. (e) Adapt the rule from (d) to write a C2 entry (referenced appropriately) that could be used to give the correct value for each different number of coins. (f) Use a spreadsheet to determine how many nickels and how many dimes (with 125 total coins) give the value $7.95.

(a) With cell A2 as 49, what entry in cell A3 (referenced to A2) could be used to ﬁll column A with the possible lengths? (b) Because the sum of the length and width must be 50, what rule would we use to ﬁnd the width if we knew the length? (c) Use your result from (b) to write an entry for cell B2 that is referenced to A2 and that could be used to ﬁll column B with the correct widths for each of the lengths in column A. (d) What entry in cell C2 (appropriately referenced) could be used to ﬁll column C with the area for any pair of lengths and widths? (e) Use a spreadsheet to determine the (whole-number) length and width for which we get the smallest area that is still greater than 550. 76. Cellular telephone subscribers The number of millions of U.S. cellular telephone subscribers y can be approximated by the equation y ⫽ 0.73t2 ⫹ 2.69t ⫹ 3.06 where t is the number of years past 1990. (Source: Cellular Telecommunications and Internet Association) Use a spreadsheet to track the predicted number of U.S. cellular telephone subscribers from 2000 to 2015. 77. Consumer price index for medical care The consumer price index (CPI) for medical care can be approximated with the equation M ⫽ 0.119t2 ⫹ 4.200t ⫹ 30.25 where t is the number of years past 1970 and M is the consumer cost, in year 1970 ⫹ t, for medical care that cost $100 in 1982. (Source: U.S. Department of Labor) (a) Use a spreadsheet to track the approximate CPI for medical care from 2000 to 2015. (b) In what year does the CPI surpass $425? (c) What governmental action might cause these estimates to be much too high?

0.6 Factoring ●

0.6 Common Factors

37

Factoring We can factor monomial factors out of a polynomial by using the Distributive Law in reverse; ab ⫹ ac ⫽ a(b ⫹ c) is an example showing that a is a monomial factor of the polynomial ab ⫹ ac. But it is also a statement of the Distributive Law (with the sides of the equation interchanged). The monomial factor of a polynomial must be a factor of each term of the polynomial, so it is frequently called a common monomial factor.

● EXAMPLE 1 Monomial Factor Factor ⫺3x2t ⫺ 3x ⫹ 9xt2. Solution 1. We can factor out 3x and obtain ⫺3x2t ⫺ 3x ⫹ 9xt2 ⫽ 3x(⫺xt ⫺ 1 ⫹ 3t2) 2. Or we can factor out ⫺3x (factoring out the negative will make the ﬁrst term of the polynomial positive) and obtain ⫺3x2t ⫺ 3x ⫹ 9xt2 ⫽ ⫺3x(xt ⫹ 1 ⫺ 3t2) If a factor is common to each term of a polynomial, we can use this procedure to factor it out, even if it is not a monomial. For example, we can factor (a ⫹ b) out of the polynomial 2x(a ⫹ b) ⫺ 3y(a ⫹ b) and get (a ⫹ b)(2x ⫺ 3y). The following example demonstrates the factoring by grouping technique.

● EXAMPLE 2 Factoring by Grouping Factor 5x ⫺ 5y ⫹ bx ⫺ by. Solution We can factor this polynomial by the use of grouping. The grouping is done so that common factors (frequently binomial factors) can be removed. We see that we can factor 5 from the ﬁrst two terms and b from the last two, which gives 5(x ⫺ y) ⫹ b(x ⫺ y) This gives us two terms with the common factor x ⫺ y, so we get (x ⫺ y)(5 ⫹ b)

Factoring Trinomials

We can use the formula for multiplying two binomials to factor certain trinomials. The formula (x ⫹ a)(x ⫹ b) ⫽ x2 ⫹ (a ⫹ b)x ⫹ ab can be used to factor trinomials such as x2 ⫺ 7x ⫹ 6.

● EXAMPLE 3 Factoring a Trinomial Factor x2 ⫺ 7x ⫹ 6. Solution If this trinomial can be factored into an expression of the form (x ⫹ a)(x ⫹ b)

38

● Chapter 0

Algebraic Concepts

then we need to ﬁnd a and b such that x2 ⫺ 7x ⫹ 6 ⫽ x2 ⫹ (a ⫹ b)x ⫹ ab That is, we need to ﬁnd a and b such that a ⫹ b ⫽ ⫺7 and ab ⫽ 6. The two numbers whose sum is ⫺7 and whose product is 6 are ⫺1 and ⫺6. Thus x2 ⫺ 7x ⫹ 6 ⫽ (x ⫺ 1)(x ⫺ 6) A similar method can be used to factor trinomials such as 9x2 ⫺ 31x ⫹ 12. Finding the proper factors for this type of trinomial may involve a fair amount of trial and error, because we must ﬁnd factors a, b, c, and d such that (ax ⫹ b)(cx ⫹ d) ⫽ acx2 ⫹ (ad ⫹ bc)x ⫹ bd Another technique of factoring is used to factor trinomials such as those we have been discussing. It is useful in factoring more complicated trinomials, such as 9x2 ⫺ 31x ⫹ 12. This procedure for factoring second-degree trinomials follows.

Factoring a Trinomial Procedure

Example

To factor a trinomial into the product of its binomial factors:

Factor 9x2 ⫺ 31x ⫹ 12.

1. Form the product of the second-degree term and the constant term.

1. 9x2 ⴢ 12 ⫽ 108x2

2. Determine whether there are any factors of the product of Step 1 that will sum to the middle term of the trinomial. (If the answer is no, the trinomial will not factor into two binomials.)

2. The factors ⫺27x and ⫺4x give a sum of ⫺31x.

3. Use the sum of these two factors to replace the middle term of the trinomial.

3. 9x2 ⫺ 31x ⫹ 12 ⫽ 9x2 ⫺ 27x ⫺ 4x ⫹ 12

4. Factor this four-term expression by grouping.

4. 9x2 ⫺ 31x ⫹ 12 ⫽ (9x2 ⫺ 27x) ⫹ (⫺4x ⫹ 12) ⫽ 9x(x ⫺ 3) ⫺ 4(x ⫺ 3) ⫽ (x ⫺ 3)(9x ⫺ 4)

In the example just completed, note that writing the middle term (⫺31x) as ⫺4x ⫺ 27x rather than as ⫺27x ⫺ 4x (as we did) will also result in the correct factorization. (Try it.)

● EXAMPLE 4 Factoring a Trinomial Factor 9x2 ⫺ 9x ⫺ 10. Solution The product of the second-degree term and the constant is ⫺90x2. Factors of ⫺90x2 that sum to ⫺9x are ⫺15x and 6x. Thus 9x2 ⫺ 9x ⫺ 10 ⫽ 9x2 ⫺ 15x ⫹ 6x ⫺ 10 ⫽ (9x2 ⫺ 15x) ⫹ (6x ⫺ 10) ⫽ 3x(3x ⫺ 5) ⫹ 2(3x ⫺ 5) ⫽ (3x ⫺ 5)(3x ⫹ 2)

0.6 Factoring ●

39

We can check this factorization by multiplying. (3x ⫺ 5)(3x ⫹ 2) ⫽ 9x2 ⫹ 6x ⫺ 15x ⫺ 10 ⫽ 9x2 ⫺ 9x ⫺ 10 Some special products that make factoring easier are as follows. Special Factorizations

The perfect-square trinomials: x2 ⫹ 2ax ⫹ a2 ⫽ (x ⫹ a)2 x2 ⫺ 2ax ⫹ a2 ⫽ (x ⫺ a)2 The difference of two squares: x2 ⫺ a2 ⫽ (x ⫹ a)(x ⫺ a)

● EXAMPLE 5 Special Factorizations (a) Factor 25x2 ⫺ 36y2.

(b) Factor 4x2 ⫹ 12x ⫹ 9

Solution (a) The binomial 25x2 ⫺ 36y2 is the difference of two squares, so we get 25x2 ⫺ 36y2 ⫽ (5x ⫺ 6y)(5x ⫹ 6y) These two factors are called binomial conjugates because they differ in only one sign. (b) Although we can use the technique we have learned to factor trinomials, the factors come quickly if we recognize that this trinomial is a perfect square. It has two square terms, and the remaining term (12x) is twice the product of the square roots of the squares (12x ⫽ 2 ⴢ 2x ⴢ 3). Thus 4x2 ⫹ 12x ⫹ 9 ⫽ (2x ⫹ 3)2 Most of the polynomials we have factored have been second-degree polynomials, or quadratic polynomials. Some polynomials that are not quadratic are in a form that can be factored in the same manner as quadratics. For example, the polynomial x4 ⫹ 4x2 ⫹ 4 can be written as a2 ⫹ 4a ⫹ 4, where a ⫽ x2.

● EXAMPLE 6 Polynomials in Quadratic Form Factor (a) x4 ⫹ 4x2 ⫹ 4 and (b) x4 ⫺ 16. Solution (a) The trinomial is in the form of a perfect square, so letting a ⫽ x2 will give us x4 ⫹ 4x2 ⫹ 4 ⫽ a2 ⫹ 4a ⫹ 4 ⫽ (a ⫹ 2)2 Thus x4 ⫹ 4x2 ⫹ 4 ⫽ (x2 ⫹ 2)2 (b) The binomial x4 ⫺ 16 can be treated as the difference of two squares, (x2)2 ⫺ 42, so x4 ⫺ 16 ⫽ (x2 ⫺ 4)(x2 ⫹ 4)

40

● Chapter 0

Algebraic Concepts

But x2 ⫺ 4 can be factored into (x ⫺ 2)(x ⫹ 2), so x4 ⫺ 16 ⫽ (x ⫺ 2)(x ⫹ 2)(x2 ⫹ 4) ●

Checkpoint

1. Factor the following. (a) 8x3 ⫺ 12x (b) 3x(x2 ⫹ 5) ⫺ 5(x2 ⫹ 5) (c) x2 ⫺ 10x ⫺ 24 (d) x2 ⫺ 5x ⫹ 6 (e) 4x2 ⫺ 20x ⫹ 25 (f) 100 ⫺ 49x2 2. Consider 10x2 ⫺ 17x ⫺ 20 and observe that (10x2)(⫺20) ⫽ ⫺200x2. (a) Find two expressions whose product is ⫺200x2 and whose sum is ⫺17x. (b) Replace ⫺17x in 10x2 ⫺ 17x ⫺ 20 with the two expressions in (a). (c) Factor (b) by grouping. 3. True or false: (a) 4x2 ⫹ 9 ⫽ (2x ⫹ 3)2 (b) x2 ⫹ x ⫺ 12 ⫽ (x ⫺ 4)(x ⫹ 3) (c) 5x5 ⫺ 20x3 ⫽ 5x3(x2 ⫺ 4) ⫽ 5x3(x ⫹ 2)(x ⫺ 2) A polynomial is said to be factored completely if all possible factorizations have been completed. For example, (2x ⫺ 4)(x ⫹ 3) is not factored completely because a 2 can still be factored out of 2x ⫺ 4. The following guidelines are used to factor polynomials completely.

Guidelines for Factoring Completely

Look for: Monomials ﬁrst. Then for: Difference of two squares. Then for: Trinomial squares. Then for: Other methods of factoring trinomials.

● EXAMPLE 7 Factoring Completely Factor completely (a) 12x2 ⫺ 36x ⫹ 27 and (b) 16x2 ⫺ 64y2. Solution (a) 12x2 ⫺ 36x ⫹ 27 ⫽ 3(4x2 ⫺ 12x ⫹ 9) Monomial ⫽ 3(2x ⫺ 3)2 Perfect Square (b) 16x2 ⫺ 64y2 ⫽ 16(x2 ⫺ 4y2) ⫽ 16(x ⫹ 2y)(x ⫺ 2y) Factoring the difference of two squares immediately would give (4x ⫹ 8y)(4x ⫺ 8y), which is not factored completely (because we could still factor 4 from 4x ⫹ 8y and 4 from 4x ⫺ 8y). ●

Checkpoint Solutions

8x3 ⫺ 12x ⫽ 4x(2x2 ⫺ 3) 3x(x2 ⫹ 5) ⫺ 5(x2 ⫹ 5) ⫽ (x2 ⫹ 5)(3x ⫺ 5) x2 ⫺ 10x ⫺ 24 ⫽ (x ⫺ 12)(x ⫹ 2) x2 ⫺ 5x ⫹ 6 ⫽ (x ⫺ 3)(x ⫺ 2) 4x2 ⫺ 20x ⫹ 25 ⫽ (2x ⫺ 5)2 100 ⫺ 49x2 ⫽ (10 ⫹ 7x)(10 ⫺ 7x) (⫺25x)(⫹8x) ⫽ ⫺200x2 and ⫺25x ⫹ 8x ⫽ ⫺17x 10x2 ⫺ 17x ⫺ 20 ⫽ 10x2 ⫺ 25x ⫹ 8x ⫺ 20 ⫽ (10x2 ⫺ 25x) ⫹ (8x ⫺ 20) ⫽ 5x(2x ⫺ 5) ⫹ 4(2x ⫺ 5) ⫽ (2x ⫺ 5)(5x ⫹ 4) 3. (a) False. 4x2 ⫹ 9 cannot be factored. In fact, sums of squares cannot be factored. (b) False. x2 ⫹ x ⫺ 12 ⫽ (x ⫹ 4)(x ⫺ 3) (c) True. 1. (a) (b) (c) (d) (e) (f) 2. (a) (b) (c)

0.6 Factoring ●

41

0.6 Exercises In Problems 1–4, factor by ﬁnding the common monomial factor. 1. 2. 3. 4. In Problems 5–8, factor by grouping. 5. 6. 7. 8. Factor each expression in Problems 9–18 as a product of binomials. 9. 10. 11. 12. 13. 14. 15. 16. 17. (a) (b) 18. (a) (b) (c) (d) In Problems 19–44, factor completely. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. In Problems 45–50 determine the missing factor. 45. 46. 47. 48. 49.

A P P L I C AT I O N S 59. Simple interest The future value of a simple-interest investment of P dollars at an annual interest rate r for t years is given by the expression . Factor this expression. 60. Reaction to medication When medicine is administered, the reaction (measured in change of blood pressure or temperature) can be modeled by (that is, described by) R⫽

cm2 m3 ⫺ 2

where c is a positive constant and m is the amount of medicine absorbed into the blood.* Factor the expression for the reaction. 61. Sensitivity to medication From the formula for reaction to medication given in Problem 60, an expression for sensitivity S can be obtained, where S ⫽ cm ⫺ m2 Factor this expression for sensitivity. 62. Volume Suppose that squares of side x are cut from four corners of an 8-by-8-inch piece of cardboard and an open-top box is formed (see Figure 0.10). The volume of the box is given by 64x ⫺ 32x2 ⫹ 4x3. Factor this expression. x x

50. Use the following factorization formulas involving cubes to factor each expression in Problems 51–58. Figure 0.10 Factorizations with Cubes Perfect cube Perfect cube

63. Consumer expenditure The consumer expenditure for a commodity is the product of its market price, p, and the number of units demanded. Suppose that for a certain commodity, the consumer expenditure is given by 10,000p ⫺ 100p2

Difference of two cubes Sum of two cubes

42

● Chapter 0

Algebraic Concepts

(a) Factor this in order to ﬁnd an expression for the number of units demanded. (b) Use (a) to ﬁnd the number of units demanded when the market price is $38.

65. Revenue Revenue R from the sale of x units of a product is found by multiplying the price by the number of items sold. (a) Factor the right side of R ⫽ 300x ⫺ x2. (b) What is the expression for the price of the item?

64. Power in a circuit Factor the following expression for the maximum power in a certain electrical circuit.

66. Poiseuille’s law The expression for the speed of blood through an artery of radius r at a distance x from the artery wall is given by r2 ⫺ (r ⫺ x)2. Factor and simplify this expression.

(R ⫹ r)2 ⫺ 2r(R ⫹ r)

0.7

Algebraic Fractions Evaluating certain limits and graphing rational functions require an understanding of algebraic fractions. The fraction 6兾8 can be reduced to 3兾4 by dividing both the numerator and the denominator by 2. In the same manner, the algebraic fraction (x ⫹ 2)(x ⫹ 1) (x ⫹ 1)(x ⫹ 3) can be reduced to x⫹2 x⫹3 by dividing both the numerator and the denominator by x ⫹ 1, if x ⫽ ⫺1.

Simplifying Fractions

We simplify algebraic fractions by factoring the numerator and denominator and then dividing both the numerator and the denominator by any common factors.*

● EXAMPLE 1 Simplifying a Fraction Simplify

3x2 ⫺ 14x ⫹ 8 if x2 ⫽ 16. x2 ⫺ 16

Solution (3x ⫺ 2)(x ⫺ 4) 3x2 ⫺ 14x ⫹ 8 ⫽ x2 ⫺ 16 (x ⫺ 4)(x ⫹ 4) 1

(3x ⫺ 2)(x ⫺ 4) ⫽ (x ⫺ 4)(x ⫹ 4) 1

3x ⫺ 2 ⫽ x⫹4

*We assume that all fractions are deﬁned.

0.7 Algebraic Fractions ●

Products of Fractions

43

We can multiply fractions by writing the product as the product of the numerators divided by the product of the denominators. For example, 4 10 2 80 ⴢ ⴢ ⫽ 5 12 5 300 which reduces to 154 . We can also ﬁnd the product by reducing the fractions before we indicate the multiplication in the numerator and denominator. For example, in 4 10 2 ⴢ ⴢ 5 12 5 we can divide the ﬁrst numerator and the second denominator by 4 and divide the second numerator and the ﬁrst denominator by 5, which yields 1

2

1

3

4 10 2 1 2 2 4 ⴢ ⴢ ⫽ ⴢ ⴢ ⫽ 5 12 5 1 3 5 15

Product of Fractions

We multiply algebraic fractions by writing the product of the numerators divided by the product of the denominators, and then reduce to lowest terms. We may also reduce prior to ﬁnding the product.

● EXAMPLE 2 Multiplying Fractions Multiply: (a)

4x2 10x y ⴢ 2 ⴢ 2 5y y 8x

(b)

⫺4x ⫹ 8 2x ⫹ 4 ⴢ 3x ⫹ 6 4x ⫹ 12

Solution 1

1

2

1

4x2 10x y 4x2 10x y 1 2x 1 x (a) ⴢ 2 ⴢ 2⫽ ⴢ 2 ⴢ 2⫽ ⴢ 2 ⴢ ⫽ 2 5y y 8x 5y y 8x 1 y 2 y 1ⴢ1

(b)

2

1

⫺4(x ⫺ 2) 2(x ⫹ 2) ⫺4x ⫹ 8 2x ⫹ 4 ⴢ ⫽ ⴢ 3x ⫹ 6 4x ⫹ 12 3(x ⫹ 2) 4(x ⫹ 3) 1

⫺1

⫺4(x ⫺ 2) 2(x ⫹ 2) ⫽ ⴢ 3(x ⫹ 2) 4(x ⫹ 3) 1

1

⫺2(x ⫺ 2) ⫽ 3(x ⫹ 3)

Quotients of Fractions

In arithmetic we learned to divide one fraction by another by inverting the divisor and multiplying. The same rule applies to division of algebraic fractions.

● EXAMPLE 3 Dividing Fractions (a) Divide

a2 b ab by 2 . c c

(b) Find

6x2 ⫺ 6 x⫺1 ⫼ 2 . x2 ⫹ 3x ⫹ 2 x ⫹ 4x ⫹ 4

44

● Chapter 0

Algebraic Concepts

Solution aⴢ1

c

1

1ⴢ1

ab a2b a2b c2 a2b c2 ac ⫼ 2 ⫽ ⴢ (a) ⫽ ⴢ ⫽ ⫽ ac c c ab c c ab 1 6x2 ⫺ 6 x⫺1 6x2 ⫺ 6 x2 ⫹ 4x ⫹ 4 (b) 2 ⫼ 2 ⫽ 2 ⴢ x ⫹ 3x ⫹ 2 x ⫹ 4x ⫹ 4 x ⫹ 3x ⫹ 2 x⫺1 6(x ⫺ 1)(x ⫹ 1) (x ⫹ 2)(x ⫹ 2) ⫽ ⴢ (x ⫹ 2)(x ⫹ 1) x⫺1 ⫽ 6(x ⫹ 2)

●

Checkpoint

Adding and Subtracting Fractions

2x2 ⫺ 4x 2x x2 x⫹3 2. Multiply: 2 ⴢ x ⫺9 3x 5x2(x ⫺ 1) 10x2 3. Divide: ⫼ 2(x ⫹ 1) (x ⫹ 1)(x ⫺ 1) 1. Reduce:

If two fractions are to be added, it is convenient that both be expressed with the same denominator. If the denominators are not the same, we can write the equivalents of each of the fractions with a common denominator. We usually use the least common denominator (LCD) when we write the equivalent fractions. The least common denominator is the lowest-degree variable expression into which all denominators will divide. If the denominators are polynomials, then the LCD is the lowest-degree polynomial into which all denominators will divide. We can ﬁnd the least common denominator as follows.

Finding the Least Common Denominator Procedure

Example

To ﬁnd the least common denominator of a set of fractions:

Find the LCD of

1. Completely factor each denominator.

1. The factored denominators are x(x ⫺ 1), (x ⫹ 1)(x ⫺ 1), and x ⴢ x.

2. Identify the different factors that appear.

2. The different factors are x, x ⫺ 1, and x ⫹ 1.

3. The LCD is the product of these different factors, with each factor used the maximum number of times it occurs in any one denominator.

3. x occurs a maximum of twice in one denominator, x ⫺ 1 occurs once, and x ⫹ 1 occurs once. Thus the LCD is x ⴢ x(x ⫺ 1)(x ⫹ 1) ⫽ x2(x ⫺ 1)(x ⫹ 1).

1 1 1 , , . x2 ⫺ x x2 ⫺ 1 x2

The procedure for combining (adding or subtracting) two or more fractions follows.

0.7 Algebraic Fractions ●

45

Adding or Subtracting Fractions Procedure

Example

To combine fractions:

Combine

1. Find the LCD of the fractions.

1. y2 ⫺ y ⫺ 20 ⫽ (y ⫺ 5)(y ⫹ 4), so the LCD is (y ⫺ 5)(y ⫹ 4).

2. Write the equivalent of each fraction with the LCD as its denominator.

2. The sum is

3. Add or subtract, as indicated, by combining like terms in the numerator over the LCD.

3. ⫽ ⫽

4. Reduce the fraction, if possible.

4. ⫽

y⫺3 y ⫺ 23 ⫹ 2 . y⫺5 y ⫺ y ⫺ 20

(y ⫺ 3)(y ⫹ 4) y ⫺ 23 ⫹ . (y ⫺ 5)(y ⫹ 4) (y ⫺ 5)(y ⫹ 4)

y2 ⫹ y ⫺ 12 ⫹ y ⫺ 23 (y ⫺ 5)(y ⫹ 4) y2 ⫹ 2y ⫺ 35 (y ⫺ 5)(y ⫹ 4) y⫹7 (y ⫹ 7)(y ⫺ 5) ⫽ , if y ⫽ 5. (y ⫺ 5)(y ⫹ 4) y⫹4

● EXAMPLE 4 Adding Fractions Add

3x 4 ⫹ . ax a2

Solution 1. The LCD is a2x. 3x x 4 a 3x2 3x 4 4a ⫽ 2ⴢ ⫹ ⴢ ⫽ 2 ⫹ 2 2. 2 ⫹ ax ax a a a x ax ax 3x2 4a 3x2 ⫹ 4a 3. 2 ⫹ 2 ⫽ ax ax a2x 4. The sum is in lowest terms.

● EXAMPLE 5 Combining Fractions Combine

y⫺3 y⫺2 ⫺ 2 . 2 (y ⫺ 5) y ⫺ 4y ⫺ 5

Solution y2 ⫺ 4y ⫺ 5 ⫽ (y ⫺ 5)(y ⫹ 1), so the LCD is (y ⫺ 5)2(y ⫹ 1). Writing the equivalent fractions and then combining them, we get y⫺3 y⫺2 (y ⫺ 3)(y ⫹ 1) (y ⫺ 2)(y ⫺ 5) ⫺ ⫽ ⫺ 2 2 (y ⫺ 5) (y ⫺ 5)(y ⫹ 1) (y ⫺ 5) (y ⫹ 1) (y ⫺ 5)(y ⫹ 1)(y ⫺ 5) ⫽

(y2 ⫺ 2y ⫺ 3) ⫺ (y2 ⫺ 7y ⫹ 10) (y ⫺ 5)2(y ⫹ 1)

⫽

y2 ⫺ 2y ⫺ 3 ⫺ y2 ⫹ 7y ⫺ 10 (y ⫺ 5)2(y ⫹ 1)

⫽

5y ⫺ 13 (y ⫺ 5)2(y ⫹ 1)

46

● Chapter 0

Algebraic Concepts

Complex Fractions

A fractional expression that contains one or more fractions in its numerator or denominator is called a complex fraction. An example of a complex fraction is 1 4 ⫹ x 3 1 3⫺ xy ac a ⫽ , with c equal to the LCD b bc of all the fractions contained in the numerator and denominator of the complex fraction. For example, all fractions contained in

We can simplify fractions of this type using the property

1 4 ⫹ x 3 1 3⫺ xy have LCD 3xy. We simplify this complex fraction by multiplying the numerator and denominator as follows: 1 1 4 4 3xya b ⫹ 3xya b 3xya ⫹ b x x xy ⫹ 12y 3 3 ⫽ ⫽ 1 1 9xy ⫺ 3 3xya3 ⫺ b 3xy(3) ⫺ 3xya b xy xy

● EXAMPLE 6 Complex Fractions Simplify

x⫺3 ⫹ x2y⫺3 so that only positive exponents remain. (xy)⫺2

Solution x⫺3

x2y⫺3

⫹ (xy)⫺2

●

Checkpoint

4. Add or subtract: 5x ⫺ 1 x⫹9 (a) ⫺ 2x ⫺ 5 2x ⫺ 5 x⫹1 x ⫹ (b) x x⫺1 y ⫺1 x 5. Simplify . y x ⫺ x y

1 x2 ⫹ 3 3 x y ⫽ , LCD ⫽ x3y3 1 (xy)2 1 x2 x3y3 a 3 ⫹ 3 b y3 ⫹ x5 x y ⫽ ⫽ xy 1 x3y3 a 2 2 b xy

0.7 Algebraic Fractions ●

Rationalizing Denominators

47

We can simplify algebraic fractions whose denominators contain sums and differences that involve square roots by rationalizing the denominators. Using the fact that (x ⫹ y)(x ⫺ y) ⫽ x2 ⫺ y2, we multiply the numerator and denominator of an algebraic fraction of this type by the conjugate of the denominator to simplify the fraction.

● EXAMPLE 7 Rationalizing Denominators 3 ⫹ 1x

Rationalize the denominators. (a)

1x ⫺ 2 1

(b)

1x ⫹ 15

Solution Multiplying 1x ⫺ 2 by 1x ⫹ 2, its conjugate, gives the difference of two squares and removes the radical from the denominator in (a). We also use the conjugate in (b).

1x ⫹ 2 1x ⫹ 2 1x ⫹ 2 ⫽ ⫽ 2 2 x⫺4 1x ⫺ 2 1x ⫹ 2 ( 1x) ⫺ (2) 3 ⫹ 1x 1x ⫺ 15 3 1x ⫺ 3 15 ⫹ x ⫺ 15x ⴢ ⫽ (b) x⫺5 1x ⫹ 15 1x ⫺ 15 (a)

●

●

Checkpoint

Checkpoint Solutions

1

ⴢ

6. Rationalize the denominator

1x

1x ⫺ 3

.

2x(x ⫺ 2) 2x2 ⫺ 4x ⫽ ⫽x⫺2 2x 2x x2 ⴢ (x ⫹ 3) x2 x⫹3 x x 2. 2 ⴢ ⫽ ⫽ ⫽ x ⫺9 3x (x ⫹ 3)(x ⫺ 3) ⴢ 3x 3(x ⫺ 3) 3x ⫺ 9 5x2(x ⫺ 1) (x ⫹ 1)(x ⫺ 1) 5x2(x ⫺ 1) 10x2 3. ⫼ ⫽ ⴢ 2(x ⫹ 1) (x ⫹ 1)(x ⫺ 1) 2(x ⫹ 1) 10x2 2 (x ⫺ 1) ⫽ 4 (5x ⫺ 1) ⫺ (x ⫹ 9) 5x ⫺ 1 x⫹9 4. (a) ⫺ ⫽ 2x ⫺ 5 2x ⫺ 5 2x ⫺ 5 5x ⫺ 1 ⫺ x ⫺ 9 ⫽ 2x ⫺ 5 2(2x ⫺ 5) 4x ⫺ 10 ⫽ ⫽ ⫽2 2x ⫺ 5 2x ⫺ 5 x x⫹1 ⫹ (b) has LCD ⫽ x(x ⫺ 1) x x⫺1 x x⫹1 x ⫹ 1 (x ⫺ 1) x x ⫹ ⴢ ⫽ ⫹ ⴢ x x x⫺1 (x ⫺ 1) x⫺1 x x2 ⫺ 1 x2 x2 ⫺ 1 ⫹ x2 ⫽ ⫹ ⫽ x(x ⫺ 1) x(x ⫺ 1) x(x ⫺ 1) 2x2 ⫺ 1 ⫽ x(x ⫺ 1) 1.

48

● Chapter 0

Algebraic Concepts

y y y a ⫺ 1b ⫺1 ⴢ xy ⫺ 1 ⴢ xy x x x xy ⫽ ⴢ ⫽ 5. xy y y y x x x a ⫺ b ⫺ ⴢ xy ⫺ ⴢ xy x x x y y y

y2 ⫺ xy y(y ⫺ x) y ⫽ ⫽ y2 ⫺ x2 (y ⫹ x)(y ⫺ x) y⫹x 1x 1x 1x ⫹ 3 x ⫹ 3 1x 6. ⫽ ⴢ ⫽ x⫺9 1x ⫺ 3 1x ⫺ 3 1x ⫹ 3 ⫽

0.7 Exercises Simplify the following fractions. 18x3y3 15a4b5 1. 2. 3 9x z 30a3b x ⫺ 3y x2 ⫺ 6x ⫹ 8 3. 4. 3x ⫺ 9y x2 ⫺ 16 2 2 x ⫺ 2x ⫹ 1 x ⫺ 5x ⫹ 6 5. 2 6. x ⫺ 4x ⫹ 3 9 ⫺ x2 3 3 2 2 6x y ⫺ 15x y x y2 ⫺ 4x3y 7. 8. 3x2y2 ⫹ 9x2y x2y ⫺ 2x2y2 In Problems 9–36, perform the indicated operations and simplify. 6x3 16x 15y4 25ac2 4ad4 ⴢ 9. 3 ⴢ 2 ⴢ 3 10. 8y 9y x 15a2c 15abc3 8x ⫺ 16 4x ⫺ 12 ⴢ 11. x⫺3 3x ⫺ 6 2x ⫺3 12. (x2 ⫺ 4) ⴢ x⫹2 x2 ⫹ 7x ⫹ 12 4x ⫹ 4 x2 ⫺ 6x ⫹ 8 ⴢ (9x ⫹ 3) 14. ⴢ 13. 2 3x ⫹ 13x ⫹ 4 x⫺4 8x2 ⫹ 8x 2 2 2 x ⫺x⫺2 18 ⫺ 2x x ⫺ 2x ⫺ 8 ⴢ 2 ⴢ 2 15. 2 2x ⫺ 8 x ⫺ 5x ⫹ 4 x ⫺ 6x ⫹ 9 x2 ⫺ 5x ⫺ 6 x2 ⫺ x ⫺ 12 x ⫺ x3 ⴢ ⴢ 2 16. 2 3 2 x ⫺ 5x ⫹ 4 x ⫺ 6x x ⫺ 2x ⫹ 1 15ac2 4a 16 4 ⫼ ⫼ 17. 18. 2 7bd 14b d x⫺2 3x ⫺ 6 y2 ⫺ 2y ⫹ 1 y2 ⫺ 4y ⫹ 3 ⫼ 19. 7y2 ⫺ 7y 35y2 2 2 6x 3x ⫹ 12x ⫼ 2 20. 2 4x y ⫺ 12xy x ⫹ x ⫺ 12 9 ⫺ x2 21. (x2 ⫺ x ⫺ 6) ⫼ 2 x ⫺ 3x 2x2 ⫹ 7x ⫹ 3 ⫼ (x ⫹ 3) 22. 4x2 ⫺ 1 x⫹2 2x ⫺ 2 23. 2 x ⫺x⫺2 x ⫺x⫺2

24. 25. 27. 29. 31. 32. 33. 34. 35. 36.

x⫹1 4 ⫺ 2 9⫺x 9 ⫺ x2 a a⫺2 2 ⫺ 26. x ⫺ a a⫺2 x⫺1 x x⫺1 2 ⫺x⫹1 ⫺ 2 28. x⫹1 x⫹1 x ⫹x 4a 5a2 b⫺1 b ⫹ ⫹ 30. 2 3x ⫹ 6 4x ⫹ 8 b ⫹ 2b 3b ⫹ 6 3x ⫺ 1 4x x⫺4 ⫹ ⫺ 2x ⫺ 4 3x ⫺ 6 5x ⫺ 10 2x ⫹ 1 5 x⫹4 ⫹ ⫺ 2 4x ⫺ 2 2x 2x ⫺ x x 4 x⫺2 ⫹ 2 ⫺ 2 2 x ⫺4 x ⫺x⫺2 x ⫹ 3x ⫹ 2 3x2 2 ⫹ 2 ⫺3 x2 ⫺ 4 x ⫺ 4x ⫹ 4 ⫺x3 ⫹ x ⫹ 2x 13 ⫺ x2 13 ⫺ x2 3x2(x ⫹ 1) ⫹ 1x3 ⫹ 1 1x3 ⫹ 1

In Problems 37–46, simplify each complex fraction. 2 4 3⫺ 3 1 1 37. 38. ⫹ 14 4 4 5 3 ⫹ y x⫹y 2y 39. 40. 1 1 1 1 ⫹ ⫹ x y 4 3y 1 2 2⫺ 1⫺ x x⫺2 41. 42. 3x 10 2x ⫺ x⫺6⫹ x⫹1 x⫹1 b 1 1a ⫺ 1x ⫺ 1 ⫹ 1a 1x ⫺ 1 43. 44. a⫺b x

Key Terms and Formulas ●

1x2 ⫹ 9 ⫺

1 ⫹9 x2 ⫺ x ⫺ 6 x⫹5 1x2 ⫹ 3 ⫺ 1x2 ⫹ 3 46. 2 x ⫹ 5x ⫹ 4 45.

13

x2

In Problems 47–50, rewrite each of the following so that only positive exponents remain, and simplify. 47. (a) (2⫺2 ⫺ 3⫺1)⫺1 (b) (2⫺1 ⫹ 3⫺1)2 2 2 ⫺1兾2 48. (a) (3 ⫹ 4 ) (b) (22 ⫹ 32)⫺1 ⫺1 ⫺2 x⫺2 ⫹ xy⫺2 2a ⫺ b 49. 50. 2 ⫺1 (ab ) (x2y)⫺2 In Problems 51 and 52, rationalize the denominator of each fraction and simplify. 1 ⫺ 1x x⫺3 51. 52. 1 ⫹ 1x x ⫺ 13 In Problems 53 and 54, rationalize the numerator of each fraction and simplify. 1x ⫹ h ⫺ 1x 19 ⫹ 2h ⫺ 3 53. 54. h h A P P L I C AT I O N S 55. Time study Workers A, B, and C can complete a job in a, b, and c hours, respectively. Working together, they can complete 1 1 1 ⫹ ⫹ a c b of the job in 1 hour. Add these fractions over a common denominator to obtain an expression for what they can do in 1 hour, working together.

(a) Combine these fractions. (b) Use the reciprocal of your answer in (a) to ﬁnd the combined focal length. Average cost A company’s average cost per unit when x units are produced is deﬁned to be Average cost ⴝ

Total cost x

Use this equation in Problems 57 and 58. 57. Suppose a company’s average costs are given by Average cost ⫽

4000 ⫹ 55 ⫹ 0.1x x

(a) Express the average-cost formula as a single fraction. (b) Write the expression that gives the company’s total costs. 58. Suppose a company’s average costs are given by Average cost ⫽

40,500 ⫹ 190 ⫹ 0.2x x

(a) Express the average-cost formula as a single fraction. (b) Write the expression that gives the company’s total costs. 59. Advertising and sales Suppose that a company’s daily sales volume attributed to an advertising campaign is given by Sales volume ⫽ 1 ⫹

3 18 ⫺ t⫹3 (t ⫹ 3)2

where t is the number of days since the campaign started. Express the sales volume as a single fraction. 60. Annuity The formula for the future value of an annuity due involves the expression

56. Focal length Two thin lenses with focal lengths p and q and separated by a distance d have their combined focal length given by the reciprocal of 1 d 1 ⫹ ⫺ p q pq

49

(1 ⫹ i)n⫹1 ⫺ 1 ⫺1 i Write this expression over a common denominator and factor the numerator to simplify.

Key Terms and Formulas Section

Key Terms

0.1

Sets and set membership Natural numbers Empty set Set equality Subset

Formulas N ⫽ 51, 2, 3, 4 . . .6 ⭋ A債B (continued)

50

● Chapter 0

Section

0.2

0.3

Algebraic Concepts

Key Terms

Formulas

Universal set Venn diagrams Set intersection Disjoint sets Set union Set complement

U

Real numbers Subsets and properties Real number line Inequalities Intervals and interval notation Closed interval Open interval Absolute value Order of operations Exponent and base Zero exponent Negative exponent Rules of exponents

0.4

Radical Radicand, index Principal nth root Fractional exponents

A傽B A傽B⫽⭋ A´B A¿

a ⱕ x ⱕ b or 3a, b4 a ⬍ x ⬍ b or (a, b)

an has base a, exponent n a0 ⫽ 1, a ⫽ 0 1 a⫺n ⫽ n a

1a n

radicand ⫽ a; index ⫽ n n 1 a ⫽ b only if bn ⫽ a and a ⱖ 0 and b ⱖ 0 when n is even n a1兾n ⫽ 1 a n n am兾n ⫽ 1 am ⫽ ( 1 a)m

Properties of radicals Rationalizing the denominator

0.5

Algebraic expression Variable Constant term Coefﬁcient; leading coefﬁcient Term Polynomial Degree Monomial Binomial Trinomial Like terms Distributive Law Binomial products Division of polynomials

anxn ⫹ p ⫹ a1x ⫹ a0

a(b ⫹ c) ⫽ ab ⫹ ac

Review Exercises ●

Section

Key Terms

0.6

Factor Common factor Factoring by grouping Special factorizations Difference of squares Perfect squares Conjugates Quadratic polynomials Factoring completely

0.7

51

Formulas

a2 ⫺ b2 ⫽ (a ⫹ b)(a ⫺ b) a2 ⫹ 2ab ⫹ b2 ⫽ (a ⫹ b)2 a2 ⫺ 2ab ⫹ b2 ⫽ (a ⫺ b)2 a ⫹ b; a ⫺ b ax2 ⫹ bx ⫹ c

Algebraic fractions Numerator Denominator Reduce Product of fractions Quotient of fractions Common denominator Least common denominator (LCD) Addition and subtraction of fractions Complex fraction Rationalize the denominator

Review Exercises

1. Is A 債 B, if A ⫽ 51, 2, 5, 76 and B ⫽ 5x: x is a positive integer, x ⱕ 86? 2. Is it true that 3 僆 5x: x ⬎ 36? 3. Are A ⫽ 51, 2, 3, 46 and B ⫽ 5x: x ⱕ 16 disjoint?

In Problems 4–7, use sets U ⴝ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A ⴝ {1, 2, 3, 9}, and B ⴝ {1, 3, 5, 6, 7, 8, 10} to ﬁnd the elements of the sets described. 4. A ´ B¿ 5. A¿ 傽 B 6. (A¿ 傽 B)¿ 7. Does (A¿ ´ B¿)¿ ⫽ A 傽 B? 8. State the property of the real numbers that is illustrated in each case. 1 1 (a) 6 ⫹ ⫽ ⫹ 6 (b) 2(3 ⴢ 4) ⫽ (2 ⴢ 3)4 3 3 1 (c) (6 ⫹ 9) ⫽ 2 ⫹ 3 3 9. Indicate whether each given expression is one or more of the following: rational, irrational, integer, natural, or meaningless. (a) p (b) 0兾6 (c) 6兾0 10. Insert the proper sign (⬍, ⫽, or ⬎) to replace each ⵧ. (a) p ⵧ 3.14 (b) ⫺100 ⵧ 0.1 (c) ⫺3 ⵧ ⫺12

For Problems 11–18, evaluate each expression. Use a calculator when necessary. 11. ƒ 5 ⫺ 11 ƒ 12. 44 ⫼ 2 ⴢ 11 ⫺ 102 (3)(2)(15) ⫺ (5)(8) 13. (⫺3)2 ⫺ (⫺1)3 14. (4)(10) 15. 2 ⫺ 33 ⫺ (2 ⫺ ƒ ⫺3 ƒ )4 ⫹ 11 16. ⫺42 ⫺ (⫺4)2 ⫹ 3 (⫺2.91)5 4 ⫹ 32 17. 18. 4 13.295 19. Write each inequality in interval notation, name the type of interval, and graph it on a real number line. (a) 0 ⱕ x ⱕ 5 (b) x ⱖ ⫺3 and x ⬍ 7 (c) (⫺4, ⬁) 傽 (⫺⬁, 0) 20. Write an inequality that represents each of the following. (a) (⫺1, 16) (b) 3⫺12, 84 (c) -3

-2

-1

0

1

52

● Chapter 0

Algebraic Concepts

21. Evaluate each of the following without a calculator. 3 0 (a) a b (b) 23 ⴢ 2⫺5 8 49 1 3 1 ⫺4 (c) 3 (d) a b a b 4 7 7 22. Use the rules of exponents to simplify each of the following with positive exponents. Assume all variables are nonzero. (a) x5 ⴢ x⫺7 (b) x8兾x⫺2 (c) (x3)3 4 ⫺2 ⫺3 ⫺2 (d) (y ) (e) (⫺y ) For Problems 23–28, rewrite each expression so that only positive exponents remain. Assume all variables are nonzero. ⫺2 ⫺(2xy2)⫺2 2 23. 24. a x2y⫺4 b ⫺2 ⫺3 2 (3x y ) 3 (⫺x4y⫺2z2)0 x⫺2 2 25. a ⫺1 b 26. 2y ⫺(x4y⫺2z2)⫺2 x⫺3y4z⫺2 ⫺1 y ⫺2 x 27. a ⫺2 ⫺3 ⫺3 b 28. a ba 2 b 3x y z 2y x 29. Find the following roots. 3 7 (a) ⫺ 1 ⫺64 (b) 14兾49 (c) 1 1.9487171 30. Write each of the following with an exponent and with the variable in the numerator. 3 4 (a) 1x (b) 1 x2 (c) 1兾 1 x 31. Write each of the following in radical form. (a) x2兾3 (b) x⫺1兾2 (c) ⫺x3兾2 32. Rationalize each of the following denominators and simplify. 5xy y (a) (b) 3 12x x 1 xy2 In Problems 33–38, use the properties of exponents to simplify so that only positive exponents remain. Assume all variables are positive. 33. x1兾2 ⴢ x1兾3 34. y⫺3兾4兾y⫺7兾4 35. x4 ⴢ x1兾4 36. 1兾(x⫺4兾3 ⴢ x⫺7兾3) 37. (x4兾5)1兾2 38. (x1兾2y2)4 In Problems 39–44, simplify variables are positive. 39. 112x3y5 3 3 41. 1 24x4y4 ⴢ 1 45x4y10 152x3y6 43. 113xy4

each expression. Assume all 40. 11250x6y9 42. 116a2b3 ⴢ 18a3b5 132x4y3 44. 16xy10

In Problems 45–62, perform the indicated operations and simplify. 45. (3x ⫹ 5) ⫺ (4x ⫹ 7) 46. x(1 ⫺ x) ⫹ x3x ⫺ (2 ⫹ x)4

47. 48. 49. 51. 53. 55. 57. 59. 60. 61. 62.

(3x3 ⫺ 4xy ⫺ 3) ⫹ (5xy ⫹ x3 (4xy3)(6x4y2) (3x ⫺ 4)(x ⫺ 1) 50. (4x ⫹ 1)(x ⫺ 2) 52. (2x ⫺ 3)2 54. (2x2 ⫹ 1)(x2 ⫹ x ⫺ 3) 56.

⫹ 4y ⫺ 1)

(3x ⫺ 1)(x ⫹ 2) (3x ⫺ 7)(2x ⫹ 1) (4x ⫹ 3)(4x ⫺ 3) (2x ⫺ 1)3 4x2y ⫺ 3x3y3 ⫺ 6x4y2 (x ⫺ y)(x2 ⫹ xy ⫹ y2) 58. 2x2y2 4 3 2 (3x ⫹ 2x ⫺ x ⫹ 4) ⫼ (x ⫹ 1) (x4 ⫺ 4x3 ⫹ 5x2 ⫹ x) ⫼ (x ⫺ 3) x4兾3(x2兾3 ⫺ x⫺1兾3) ( 1x ⫹ 1a ⫺ x)( 1x ⫺ 1a ⫺ x)

In Problems 63–73, factor each expression completely. 63. 2x4 ⫺ x3 64. 4(x2 ⫹ 1)2 ⫺ 2(x2 ⫹ 1)3 65. 4x2 ⫺ 4x ⫹ 1 66. 16 ⫺ 9x2 67. 2x4 ⫺ 8x2 68. x2 ⫺ 4x ⫺ 21 69. 3x2 ⫺ x ⫺ 2 70. x2 ⫺ 5x ⫹ 6 71. x2 ⫺ 10x ⫺ 24 72. 12x2 ⫺ 23x ⫺ 24 73. 16x4 ⫺ 72x2 ⫹ 81 74. Factor as indicated: x⫺2兾3 ⫹ x⫺4兾3 ⫽ x⫺4兾3(?) 75. Reduce each of the following to lowest terms. 4x2y3 ⫺ 6x3y4 2x (a) (b) 2x ⫹ 4 2x2y2 ⫺ 3xy3 In Problems 76–82, perform the indicated operations and simplify. x2 ⫺ 4x x4 ⫺ 16 76. 2 ⴢ x ⫹ 4 x4 ⫺ 16x2 x2 ⫹ 6x ⫹ 9 x2 ⫹ 4x ⫹ 3 ⫼ 2 77. 2 x ⫺ 7x ⫹ 12 x ⫺ 3x ⫺ 4 4 3 3 3 x ⫺ 2x x ⫺ 4x 1 ⫼ 2 ⫺ 2 78. 79. 1 ⫹ 3x2 ⫺ x ⫺ 2 9x ⫺ 4 2x 6x 1 x⫺2 x⫹2 x2 ⫹ 4 ⫺ ⫺ 2 ⫹1 80. 81. 2 x⫺2 4 x ⫺x x ⫺ 2x ⫹ 1 x⫺1 x 1 ⫺ 2 ⫹ 82. 2 x ⫺x⫺2 x ⫺ 2x ⫺ 3 x⫺2 In Problems 83 and 84, simplify each complex fraction. x⫺1 x⫺1⫺ x x⫺2 ⫺ x⫺1 83. 84. ⫺2 1 x ⫹ x⫺1 ⫹1 x⫺1 3x ⫺ 3 85. Rationalize the denominator of and simplify. 1x ⫺ 1 1x ⫺ 1x ⫺ 4 86. Rationalize the numerator of and 2 simplify.

Review Exercises ●

A P P L I C AT I O N S 87. Job effectiveness factors In an attempt to determine some off-the-job factors that might be indicators of onthe-job effectiveness, a company made a study of 200 of its employees. It was interested in whether the employees had been recognized for superior work by their supervisors within the past year, whether they were involved in community activities, and whether they followed a regular exercise plan. The company found the following. 30 answered “yes” to all three 50 were recognized and they exercised 52 were recognized and were involved in the community 77 were recognized 37 were involved in the community but did not exercise 95 were recognized or were involved in the community 95 answered “no” to all three (a) Draw a Venn diagram that represents this information. (b) How many exercised only? (c) How many exercised or were involved in the community? 88. Health insurance coverage The percent of the U.S. population covered by an employment-based health insurance plan can be approximated by the expression ⫺0.75x ⫹ 63.8 where x is the number of years past 2000. (Source: U.S. Census Bureau) Use this expression to estimate the percent covered by such a plan in the year 2015. 89. Poiseuille’s law The expression for the speed of blood through an artery of radius r at a distance x from the artery wall is given by r2 ⫺ (r ⫺ x)2. Evaluate this expression when r ⫽ 5 and x ⫽ 2. 90. Future value If an individual makes monthly deposits of $100 in an account that earns 9% compounded monthly, then the future value S of the account after n months is given by the formula S ⫽ $100 c

(1.0075)n ⫺ 1 d 0.0075

(a) Find the future value after 36 months (3 years). (b) Find the future value after 20 years. 91. Health care According to the American Hospital Association, the trends since 1975 indicate that the

53

number of hospital beds B (in thousands) is related to the number of hospitals H by the formula B ⫽ 176.896(1.00029)H (a) In 2004 the number of hospitals had fallen to 5759. How many beds does the equation estimate were available? (b) If the number of hospitals reaches 5000, what does the equation predict for the number of beds? 92. Severe weather ice makers Thunderstorms severe enough to produce hail develop when an upper-level low (a pool of cold air high in the atmosphere) moves through a region where there is warm, moist air at the surface. These storms create an updraft that draws the moist air into subfreezing air above 10,000 feet. The strength of the updraft, as measured by its speed s (in mph), affects the size of the hail according to h ⫽ 0.000595s1.922

or s ⫽ 47.7h0.519

where h is the diameter of the hail (in inches). (Source: National Weather Service) (a) What size hail is produced by an updraft of 50 mph? (b) When a storm produces softball-sized hail (about 4.5 inches in diameter), how fast is the updraft? 93. Loan payment Suppose you borrow $10,000 for n months to buy a car at an interest rate of 7.8% compounded monthly. The size of each month’s payment R is given by the formula R ⫽ $10,000 c

0.0065 d 1 ⫺ (1.0065)⫺n

(a) Rewrite the expression on the right-hand side of this formula as a fraction with only positive exponents. (b) Find the monthly payment for a 48-month loan. Use both the original formula and your result from (a). (Both formulas should give the same payment.) 94. Environment Suppose that in a study of water birds, the relationship between the number of acres of wetlands A and the number of species of birds S found in the wetlands area was given by S ⫽ kA1兾3 where k is a constant. (a) Express this formula using radical notation. (b) If the area is expanded by a factor of 2.25 from 20,000 acres to 45,000 acres, ﬁnd the expected increase in the number of species (as a multiple of the number of species on the 20,000 acres).

54

● Chapter 0

Algebraic Concepts

95. Proﬁt Suppose that the total cost of producing and selling x units of a product is 300 ⫹ 4x and the total revenue from the sale of x units is 30x ⫺ 0.001x2. Find the difference of these expressions (total revenue minus total cost) to ﬁnd the proﬁt from the production and sale of x units of the product. 96. Business loss Suppose that a commercial building costs $1,450,000. After it is placed into service, it loses 0.25% of its original value in each of the x months it is in service. Write an expression for the value of the building after x months. 97. Revenue The revenue for a boat tour is 600 ⫺ 13x ⫺ 0.5x2, where x is the number of passengers above the minimum of 50. This expression will factor into two binomials, with one representing the total number of passengers. Factor this expression.

C⫽

1,200,000 ⫺ 12,000 100 ⫺ p

(a) Express the right-hand side of this formula as a single fraction. (b) Find the cost if p ⫽ 0. Write a sentence that explains the meaning of what you found. (c) Find the cost of removing 98% of the pollution. (d) What happens to this formula when p ⫽ 100? Explain why you think this happens. 99. Average cost The average cost of producing a product is given by the expression 1200 ⫹ 56x ⫹

8000 x

Write this expression with all terms over a common denominator.

98. Pollution: cost-beneﬁt Suppose that the cost C (in dollars) of removing p% of the pollution from the waste water of a manufacturing process is given by

0

Chapter Test

1. Let U ⫽ 51, 2, 3, 4, 5, 6, 7, 8, 96, A ⫽ 5x: x is even and x ⱖ 56, and B ⫽ 51, 2, 5, 7, 8, 96. Complete the following. (a) Find A ´ B¿. (b) Find a two-element set that is disjoint from B. (c) Find a nonempty subset of A that is not equal to A. 2. Evaluate (4 ⫺ 23)2 ⫺ 34 ⴢ 015 ⫹ 12 ⫼ 3 ⫹ 1. 3. Use deﬁnitions and properties of exponents to complete the following. (a) x4 ⴢ x4 ⫽ x? (b) x0 ⫽ ?, if x ⫽ 0 ? (c) 1x ⫽ x (d) (x⫺5)2 ⫽ x? 27 ⫺3 ? (e) a ⫼ a ⫽ a (f) x1兾2 ⴢ x1兾3 ⫽ x? 1 1 1 (g) 3 ⫽ ? (h) 3 ⫽ x? 2 x x 1x 4. Write each of the following as radicals. (a) x1兾5 (b) x⫺3兾4 5. Simplify each of the following so that only positive exponents remain. x⫺8y2 ⫺3 (a) x⫺5 (b) a ⫺1 b x 6. Simplify the following radical expressions, and rationalize any denominators. x 1 ⫺ 1x (a) (b) 124a2b 1a3b4 (c) 15x 1 ⫹ 1x

7. Given the expression 2x3 ⫺ 7x5 ⫺ 5x ⫺ 8, complete the following. (a) Find the degree. (b) Find the constant term. (c) Find the coefﬁcient of x. 8. Express (⫺2, ⬁) 傽 (⫺⬁, 34 using interval notation, and graph it. 9. Completely factor each of the following. (a) 8x3 ⫺ 2x2 (b) x2 ⫺ 10x ⫹ 24 2 (c) 6x ⫺ 13x ⫹ 6 (d) 2x3 ⫺ 32x5 10. Identify the quadratic polynomial from among (a)–(c), and evaluate it using x ⫽ ⫺3. (a) 2x2 ⫺ 3x3 ⫹ 7 (b) x2 ⫹ 3兾x ⫹ 11 2 (c) 4 ⫺ x ⫺ x 11. Use long division to ﬁnd (2x3 ⫹ x2 ⫺ 7) ⫼ (x2 ⫺ 1). 12. Perform the indicated operations and simplify. (a) 4y ⫺ 5(9 ⫺ 3y) (b) ⫺3t2(2t4 ⫺ 3t7) 2 (c) (4x ⫺ 1)(x ⫺ 5x ⫹ 2) (d) (6x ⫺ 1)(2 ⫺ 3x) x6 x⫺3 ⴢ (e) (2m ⫺ 7)2 (f) 2 x ⫺9 3x2 4 3 x 9x 4 x⫺2 ⫺ (g) 2 ⫼ 6 (h) 3 x x⫺8 x⫺8 3 x⫺1 ⫺ 2 (i) 2 x ⫺ 2x ⫺ 3 x ⫺ 3x

Chapter Test ●

1 1 ⫺ x y . 13. Simplify 1 ⫹y x 14. In a nutrition survey of 320 students, the following information was obtained. 145 ate breakfast 270 ate lunch 280 ate dinner 125 ate breakfast and lunch 110 ate breakfast and dinner 230 ate lunch and dinner 90 ate all three meals

55

(a) Make a Venn diagram that represents this information. (b) How many students in the survey ate only breakfast? (c) How many students skipped breakfast? 15. If $1000 is invested for x years at 8% compounded quarterly, the future value of the investment is given by S ⫽ 1000a1 ⫹

0.08 4x b 4

What will be the future value of this investment in 20 years?

Extended Applications & Group Projects I. Campaign Management A politician is trying to win election to the city council, and as his campaign manager, you need to decide how to promote the candidate. There are three ways you can do so: you can send glossy, full-color pamphlets to registered voters of the city, you can run a commercial during the television news on a local cable network, and/or you can buy a full-page ad in the newspaper. Two hundred ﬁfty thousand voters live in the city, and 36% of them read the newspaper. Fifty thousand voters watch the local cable network news, and 30% of them also read the newspaper. You also know that the television commercial would cost $40,000, the newspaper ad $27,000, and the pamphlets mailed to voters 90 cents each, including printing and bulk-rate postage. Suppose that the success of the candidate depends on your campaign reaching at least 125,000 voters and that, because your budget is limited, you must achieve this goal at a minimum cost. What would be your plan and the cost of that plan? If you need help devising a method of solution for this problem, try answering the following questions ﬁrst. 1. How many voters in the city read the newspaper but do not watch the local cable television news? 2. How many voters read the newspaper or watch the local cable television news, or both? 3. Complete the following chart by indicating the number of voters reached by each promotional option, the total cost, and the cost per voter reached. Number of Voters Reached

Total Cost

Pamphlet Television Newspaper

4. Now explain your plan and the cost of that plan.

56

Cost per Voter Reached

Extended Applications & Group Projects ●

II. Pricing for Maximum Profit Overnight Cleaners specializes in cleaning professional ofﬁces. The company is expanding into a new city and believes that its staff is capable of handling 300 ofﬁce units at a weekly cost per unit of $116 for labor and supplies. Preliminary pricing surveys indicate that if Overnight’s weekly charge is $200 per unit, then it would have clients for 300 units. However, for each $10 price increase, it could expect to have 10 fewer units. By using a spreadsheet to investigate possible prices per unit and corresponding numbers of units, determine what weekly price Overnight Cleaners should charge in order to have the greatest proﬁt. The following steps will aid in creating a spreadsheet that can determine the price that gives maximum proﬁt. (a) Begin with columns for price per unit and number of units, and create a plan by hand to determine what your spreadsheet should look like. Use the information in the problem statement to determine the starting values for price per unit and number of units. Note that a price increase of $10 resulting in a reduction of 10 units to clean can be equivalently expressed as a $1 increase resulting in a 1-unit reduction. With this observation, determine the next three entries in the columns for price per unit and number of units. Write the rules used for each. (b) Proﬁt is found by subtracting expenses (or costs) from income. In the hand-done plan begun in (a), add a column or columns that would allow you to use information about price per unit and number of units to ﬁnd proﬁt. In your hand-done plan, complete entries for each new column (as you did for price per unit and number of units). Write the rule(s) you used. (At this point, you should have a hand-done plan with at least four entries in each column and a rule for ﬁnding more entries in each column.) (c) Use the starting values from (a) and the rules from (a) and (b) to create a computer spreadsheet (with appropriate cell references) that will duplicate and extend your handdone plan. (d) Fill the computer spreadsheet from (c) and ﬁnd the price per unit that gives Overnight Cleaners its maximum proﬁt. Find its maximum proﬁt. (Print the spreadsheet.)

57

1 Linear Equations and Functions A wide variety of problems from business, the social sciences, and the life sciences may be solved using equations. Managers and economists use equations and their graphs to study costs, sales, national consumption, or supply and demand. Social scientists may plot demographic data or try to develop equations that predict population growth, voting behavior, or learning and retention rates. Life scientists use equations to model the flow of blood or the conduction of nerve impulses and to test theories or develop new models by using experimental data. Numerous applications of mathematics are given throughout the text, but most chapters contain special sections emphasizing business and economics applications. In particular, this chapter introduces two important applications that will be expanded and used throughout the text as increased mathematical skills permit: supply and demand as functions of price (market analysis); and total cost, total revenue, and profit as functions of the quantity produced and sold (theory of the firm). The topics and applications discussed in this chapter include the following.

Sections

Applications

1.1 Solutions of Linear Equations and Inequalities in One Variable

Future value of an investment, voting, normal height for a given age, profit

1.2 Functions Function notation Operations with functions

Income taxes, stock market, mortgage payments

1.3 Linear Functions Graphs Slopes Equations

Depreciation, U.S. banks, pricing

1.4 Graphs and Graphing Utilities Graphical solutions of linear equations

Water purity, women in the work force

1.5 Solutions of Systems of Linear Equations Systems of linear equations in three variables

Investment mix, medicine concentrations, college enrollment

1.6 Applications of Functions in Business and Economics

Total cost, total revenue, profit, break-even analysis, demand and supply, market equilibrium

Chapter Warm-up Prerequisite Problem Type

For Section

Evaluate: (a) 2(⫺1)3 ⫺ 3(⫺1)2 ⫹ 1 (b) 3(⫺3) ⫺ 1 (c) 14(10) ⫺ 0.02(102) 3⫺1 ⫺1 ⫺ 3 (d) (e) 4 ⫺ (⫺2) 2 ⫺ (⫺2) 3(⫺8) 23 2 (f ) (g) 2¢ ≤ ⫺ 5¢ ≤ 4 9 9

1.1 1.2 1.3 1.5 1.6

Graph x ⱕ 3.

1.1

Answer

Section for Review

(a) ⫺4 (b) ⫺10 (c) 138 1 (d) 3

0.2 Signed numbers

(f) ⫺6 –2

0

(e) ⫺1 (g) 4

2

0.2 Inequalities

4 x

(a)

1 is undeﬁned for which real x number(s)?

1.1, 1.2

(b) 1x ⫺ 4 is a real number for which values of x? 1.1, 1.3 1.5, 1.6

Simplify: (a) 4(⫺c)2 ⫺ 3(⫺c) ⫹ 1 (b) 33(x ⫹ h) ⫺ 14 ⫺ 33x ⫺ 14 3x (c) 12a ⫹ 3b 4 (d) 9x ⫺ (300 ⫹ 2x) 1 (e) ⫺ 3x ⫺ (⫺1)4 5 (f) 2(2y ⫹ 3) ⫹ 3y

1.1, 1.2 1.3, 1.5 1.6

3x 1 and 2x ⫹ 10 x⫹5

0.2 Real numbers 0.4 Radicals

(b) x ⱖ 4

Identify the coefﬁcient of x and the constant term for: (a) ⫺9x ⫹ 2 x (b) 2 (c) x ⫺ 300

Find the LCD of

(a) Undeﬁned for x⫽0

Coeff. (a) ⫺9 1 (b) 2 (c) 1

Const. 2

0.5 Algebraic expressions

0 ⫺300

(a) 4c2 ⫹ 3c ⫹ 1 (b) 3h

0.5 Algebraic expressions

(c) 9x ⫹ 36 (d) 7x ⫺ 300 1 1 (e) ⫺ x ⫺ 5 5 (f) 7y ⫹ 6

1.1

2x ⫹ 10

0.7 Algebraic fractions

59

60

● Chapter 1

Linear Equations and Functions

1.1 OBJECTIVES ●

●

●

To solve linear equations in one variable. To solve applied problems by using linear equations. To solve linear inequalities in one variable.

Solutions of Linear Equations and Inequalities in One Variable ] Application Preview Using data from 1952–2004, the percent p of the eligible U.S. population voting in presidential elections has been estimated to be p ⴝ 63.20 ⴚ 0.26x where x is the number of years past 1950. (Source: Federal Election Commission) To find the election year in which the percent voting was equal to 55.4%, we solve the equation 55.4 ⴝ 63.20 ⴚ 0.26x . (See Example 4.) We will discuss solutions of linear equations and inequalities in this section.

Equations

An equation is a statement that two quantities or algebraic expressions are equal. The two quantities on either side of the equals sign are called members of the equation. For example, 2 ⫹ 2 ⫽ 4 is an equation with members 2 ⫹ 2 and 4; 3x ⫺ 2 ⫽ 7 is an equation with 3x ⫺ 2 as its left member and 7 as its right member. Note that the equation 7 ⫽ 3x ⫺ 2 is the same statement as 3x ⫺ 2 ⫽ 7. An equation such as 3x ⫺ 2 ⫽ 7 is known as an equation in one variable. The x in this case is called a variable because its value determines whether the equation is true. For example, 3x ⫺ 2 ⫽ 7 is true only for x ⫽ 3. Finding the value(s) of the variable(s) that make the equation true—that is, ﬁnding the solutions—is called solving the equation. The set of solutions of an equation is called a solution set of the equation. The variable in an equation is sometimes called the unknown. Some equations involving variables are true only for certain values of the variables, whereas others are true for all values. Equations that are true for all values of the variables are called identities. The equation 2(x ⫺ 1) ⫽ 2x ⫺ 2 is an example of an identity. Equations that are true only for certain values of the variables are called conditional equations or simply equations. Two equations are said to be equivalent if they have exactly the same solution set. For example, 4x ⫺ 12 ⫽ 16 4x ⫽ 28 x⫽7 are equivalent equations because they all have the same solution, namely 7. We can often solve a complicated linear equation by ﬁnding an equivalent equation whose solution is easily found. We use the following properties of equality to reduce an equation to a simple equivalent equation.

Properties of Equality Substitution Property The equation formed by substituting one expression for an equal expression is equivalent to the original equation. Addition Property The equation formed by adding the same quantity to both sides of an equation is equivalent to the original equation.

Examples 3(x ⫺ 3) ⫺ 12(4x ⫺ 18) ⫽ 4 is equivalent to 3x ⫺ 9 ⫺ 2x ⫹ 9 ⫽ 4 and to x ⫽ 4. We say the solution set is {4}, or the solution is 4. x ⫺ 4 ⫽ 6 is equivalent to x ⫺ 4 ⫹ 4 ⫽ 6 ⫹ 4, or to x ⫽ 10. x ⫹ 5 ⫽ 12 is equivalent to x ⫹ 5 ⫹ (⫺5) ⫽ 12 ⫹ (⫺5), or to x ⫽ 7.

1.1 Solutions of Linear Equations and Inequalities in One Variable ●

Properties of Equality

Examples

Multiplication Property The equation formed by multiplying both sides of an equation by the same nonzero quantity is equivalent to the original equation.

Solving Linear Equations

61

⫽ 6 is equivalent to 3(13x) ⫽ 3(6), or to x ⫽ 18. 5x ⫽ 20 is equivalent to (5x)>5 ⫽ 20>5, or to x ⫽ 4. (Dividing both sides by 5 is equivalent to multiplying both sides by 15 .) 1 3x

If an equation contains one variable and if the variable occurs to the ﬁrst degree, the equation is called a linear equation in one variable. The three properties above permit us to reduce any linear equation in one variable (also called an unknown) to an equivalent equation whose solution is obvious. We may solve linear equations in one variable by using the following procedure.

Solving a Linear Equation Procedure

Example

To solve a linear equation in one variable:

Solve

1. If the equation contains fractions, multiply both sides by the least common denominator (LCD) of the fractions.

1. LCD is 12. 3x x⫺1 12a ⫹ 3b ⫽ 12a b 4 3

3x x⫺1 ⫹3⫽ . 4 3

2. 9x ⫹ 36 ⫽ 4x ⫺ 4

2. Remove any parentheses in the equation. 3. Perform any additions or subtractions to get all terms containing the variable on one side and all other terms on the other side.

3. 9x ⫹ 36 ⫺ 4x ⫽ 4x ⫺ 4 ⫺ 4x 5x ⫹ 36 ⫽ ⫺4 5x ⫹ 36 ⫺ 36 ⫽ ⫺4 ⫺ 36 5x ⫽ ⫺40

4. Divide both sides of the equation by the coefﬁcient of the variable.

4.

5x ⫺40 ⫽ 5 5

5. Check the solution by substitution in the original equation.

5.

3(⫺8) ⫺8 ⫺ 1 ⫹3ⱨ 4 3

gives

x ⫽ ⫺8 gives ⫺3 ⫽ ⫺3 ✔

● EXAMPLE 1 Linear Equations (a) Solve for z:

2z ⫽ ⫺6 3

(b) Solve for x:

Solution (a) Multiply both sides by 3. 3a

2z b ⫽ 3(⫺6) 3

3x ⫹ 1 x ⫽ ⫺3 2 3

gives

2z ⫽ ⫺18

62

● Chapter 1

Linear Equations and Functions

Divide both sides by 2. 2z ⫺18 ⫽ 2 2

gives

z ⫽ ⫺9

2(⫺9) ⱨ ⫺6 gives ⫺6 ⫽ ⫺6 ✔ 3 3x ⫹ 1 x (b) ⫽ ⫺3 2 3 3x ⫹ 1 x 6a b ⫽ 6 a ⫺ 3b Multiply both sides by the LCD, 6. 2 3 x 3(3x ⫹ 1) ⫽ 6 a ⫺ 3b Simplify the fraction on the left side. 3 9x ⫹ 3 ⫽ 2x ⫺ 18 Distribute to remove parentheses. 7x ⫽ ⫺21 Add (⫺2x) ⫹ (⫺3) to both sides. x ⫽ ⫺3 Divide both sides by 7. 3(⫺3) ⫹ 1 ⫺3 ⱨ ⫺ 3 gives ⫺4 ⫽ ⫺4 ✔ Check: 2 3 Check:

● EXAMPLE 2 Future Value of an Investment The future value of a simple interest investment is given by S ⫽ P ⫹ Prt, where P is the principal invested, r is the annual interest rate (as a decimal), and t is the time in years. At what simple interest rate r must P ⫽ 1500 dollars be invested so that the future value is $2940 after 8 years? Solution Entering the values S ⫽ 2940, P ⫽ 1500, and t ⫽ 8 into S ⫽ P ⫹ Prt gives 2940 ⫽ 1500 ⫹ 1500(r)(8) Hence we solve the equation 2940 ⫽ 1500 ⫹ 12,000r 1440 ⫽ 12,000r 1440 ⫽r 12,000 0.12 ⫽ r

Subtract 1500 from both sides. Divide both sides by 12,000.

Thus the interest rate is 0.12, or 12%. Checking, we see that 2940 ⫽ 1500 ⫹ 1500(0.12)(8). ✔

Fractional Equations

A fractional equation is an equation that contains a variable in a denominator. It is solved by ﬁrst multiplying both sides of the equation by the least common denominator (LCD) of the fractions in the equation. Some fractional equations lead to linear equations. Note that the solution to any fractional equation must be checked in the original equation, because multiplying both sides of a fractional equation by a variable expression may result in an equation that is not equivalent to the original equation. If a solution to the fraction-free linear equation makes a denominator of the original equation equal to zero, that value cannot be a solution to the original equation. Some fractional equations have no solutions.

1.1 Solutions of Linear Equations and Inequalities in One Variable ●

63

● EXAMPLE 3 Solving Fractional Equations Solve for x: (a)

3x 1 ⫽1⫹ 2x ⫹ 10 x⫹5

(b)

2x ⫺ 1 5 ⫽4⫹ x⫺3 x⫺3

Solution (a) First multiply each term on both sides by the LCD, 2x ⫹ 10. Then simplify and solve. 3x 1 (2x ⫹ 10)a b ⫽ (2x ⫹ 10)(1) ⫹ (2x ⫹ 10)a b 2x ⫹ 10 x⫹5 3x ⫽ (2x ⫹ 10) ⫹ 2 gives x ⫽ 12 3(12) 1 36 18 ⱨ 1⫹ gives ⫽ Check: ✔ 2(12) ⫹ 10 12 ⫹ 5 34 17 (b) First multiply each term on both sides by the LCD, x ⫺ 3. Then simplify. 2x ⫺ 1 5 (x ⫺ 3)a b ⫽ (x ⫺ 3)(4) ⫹ (x ⫺ 3)a b x⫺3 x⫺3 2x ⫺ 1 ⫽ (4x ⫺ 12) ⫹ 5 or 2x ⫺ 1 ⫽ 4x ⫺ 7 Add (⫺4x) ⫹ 1 to both sides. ⫺2x ⫽ ⫺6 gives x ⫽ 3 Check: The value x ⫽ 3 gives undeﬁned expressions because the denominators equal 0 when x ⫽ 3. Hence the equation has no solution. ✔ ●

Checkpoint

1. Solve the following for x, and check. 5(x ⫺ 3) x (a) 4(x ⫺ 3) ⫽ 10x ⫺ 12 (b) ⫺x⫽1⫺ 6 9

(c)

2x x ⫽2⫺ 3x ⫺ 6 x⫺2

When using an equation to describe (model) a set of data, it is sometimes useful to let a variable represent the number of years past a given year. This is called aligning the data. Consider the following example. ] EXAMPLE 4 Voting (Application Preview) Using data from 1952–2004, the percent p of the eligible U.S. population voting in presidential elections has been estimated to be p ⫽ 63.20 ⫺ 0.26x where x is the number of years past 1950. (Source: Federal Election Commission) According to this model, in what election year is the percent voting equal to 55.4%? Solution To answer this question, we solve 55.4 ⫽ 63.20 ⫺ 0.26x ⫺7.8 ⫽ ⫺0.26x 30 ⫽ x Checking reveals that 55.4 ⫽ 63.20 ⫺ 0.26(30). ✔ Thus the percent of eligible voters who voted is estimated to be 55.4% in the presidential election year 1950 ⫹ 30 ⫽ 1980.

64

● Chapter 1

Linear Equations and Functions

Linear Equations with Two Variables

The steps used to solve linear equations in one variable can also be used to solve linear equations in more than one variable for one of the variables in terms of the other. Solving an equation such as the one in the following example is important when using a graphing utility.

● EXAMPLE 5 Solving an Equation for One of Two Variables Solve 4x ⫹ 3y ⫽ 12 for y. Solution No fractions or parentheses are present, so we subtract 4x from both sides to get only the term that contains y on one side. 3y ⫽ ⫺4x ⫹ 12 Dividing both sides by 3 gives the solution. 4 y⫽⫺ x⫹4 3 ⫺4 x ⫹ 4b ⱨ 12 3 4x ⫹ (⫺4x ⫹ 12) ⫽ 12 ✔

Check: 4x ⫹ 3a

●

Checkpoint

2. Solve for y: y ⫺ 4 ⫽ ⫺4(x ⫹ 2)

● EXAMPLE 6 Profit Suppose that the relationship between a ﬁrm’s proﬁt P and the number x of items sold can be described by the equation 5x ⫺ 4P ⫽ 1200 (a) How many units must be produced and sold for the ﬁrm to make a proﬁt of $150? (b) Solve this equation for P in terms of x. (c) Find the proﬁt when 240 units are sold. Solution (a) 5x ⫺ 4(150) ⫽ 1200 5x ⫺ 600 ⫽ 1200 5x ⫽ 1800 x ⫽ 360 units Check: 5(360) ⫺ 4(150) ⫽ 1800 ⫺ 600 ⫽ 1200 ✔ (b)

5x ⫺ 4P ⫽ 1200 5x ⫺ 1200 ⫽ 4P 5x 1200 5 P⫽ ⫺ ⫽ x ⫺ 300 4 4 4 5 (c) P ⫽ x ⫺ 300 4 5 P ⫽ (240) ⫺ 300 ⫽ 0 4 Because P ⫽ 0 when x ⫽ 240, we know that proﬁt is $0 when 240 units are sold, and we say that the ﬁrm breaks even when 240 units are sold.

1.1 Solutions of Linear Equations and Inequalities in One Variable ●

Stated Problems

65

With an applied problem, it is frequently necessary to convert the problem from its stated form into one or more equations from which the problem’s solution can be found. The following guidelines may be useful in solving stated problems.

Guidelines for Solving Stated Problems 1. Begin by reading the problem carefully to determine what you are to find. Use variables to represent the quantities to be found. 2. Reread the problem and use your variables to translate given information into algebraic expressions. Often, drawing a figure is helpful. 3. Use the algebraic expressions and the problem statement to formulate an equation (or equations). 4. Solve the equation(s). 5. Check the solution in the problem, not just in your equation or equations. The answer should satisfy the conditions.

● EXAMPLE 7 Investment Mix Jill Bell has $90,000 to invest. She has chosen one relatively safe investment fund that has an annual yield of 10% and another, riskier one that has a 15% annual yield. How much should she invest in each fund if she would like to earn $10,000 in one year from her investments? Solution We want to ﬁnd the amount of each investment, so we begin as follows: Let x ⫽ the amount invested at 10%, then 90,000 ⫺ x ⫽ the amount invested at 15% (because the two investments total $90,000) If P is the amount of an investment and r is the annual rate of yield (expressed as a decimal), then the annual earnings I ⫽ Pr. Using this relationship, we can summarize the information about these two investments in a table. 10% investment 15% investment Total investment

P

r

I

x 90,000 ⫺ x 90,000

0.10 0.15

0.10x 0.15(90,000 ⫺ x) 10,000

The column under I shows that the sum of the earnings is 0.10x ⫹ 0.15(90,000 ⫺ x) ⫽ 10,000 We solve this as follows. 0.10x ⫹ 13,500 ⫺ 0.15x ⫽ 10,000 ⫺0.05x ⫽ ⫺3500 or x ⫽ 70,000 Thus the amount invested at 10% is $70,000, and the amount invested at 15% is 90,000 ⫺ 70,000 ⫽ 20,000. Check: To check, we return to the problem and note that 10% of $70,000 plus 15% of $20,000 gives a yield of $7000 ⫹ $3000 ⫽ $10,000. ✔

66

● Chapter 1

Linear Equations and Functions

Linear Inequalities

An inequality is a statement that one quantity is greater than (or less than) another quantity. For example, if x represents the number of items a ﬁrm produces and sells, x must be a nonnegative quantity. Thus x is greater than or equal to zero, which is written x ⱖ 0. The inequality 3x ⫺ 2 ⬎ 2x ⫹ 1 is a ﬁrst-degree (linear) inequality that states that the left member is greater than the right member. Certain values of the variable will satisfy the inequality. These values form the solution set of the inequality. For example, 4 is in the solution set of 3x ⫺ 2 ⬎ 2x ⫹ 1 because 3 ⴢ 4 ⫺ 2 ⬎ 2 ⴢ 4 ⫹ 1. On the other hand, 2 is not in the solution set because 3 ⴢ 2 ⫺ 2 ⬎ 2 ⴢ 2 ⫹ 1. Solving an inequality means ﬁnding its solution set, and two inequalities are equivalent if they have the same solution set. As with equations, we ﬁnd the solutions to inequalities by ﬁnding equivalent inequalities from which the solutions can be easily seen. We use the following properties to reduce an inequality to a simple equivalent inequality.

Inequalities Properties

Examples

Substitution Property The inequality formed by substituting one expression for an equal expression is equivalent to the original inequality. Addition Property The inequality formed by adding the same quantity to both sides of an inequality is equivalent to the original inequality.

5x ⫺ 4x ⬍ 6 x⬍6 The solution set is 5x: x ⬍ 66 .

2x ⫺ 4 ⬎ x ⫹ 6 2x ⫺ 4 ⫹ 4 ⬎ x ⫹ 6 ⫹ 4 2x ⬎ x ⫹ 10 2x ⫹ (⫺x) ⬎ x ⫹ 10 ⫹ (⫺x) x ⬎ 10

Multiplication Property I The inequality formed by multiplying both sides of an inequality by the same positive quantity is equivalent to the original inequality.

Multiplication Property II The inequality formed by multiplying both sides of an inequality by the same negative number and reversing the direction of the inequality symbol is equivalent to the original inequality.

1 x⬎8 2 1 x(2) ⬎ 8(2) 2 x ⬎ 16

⫺x ⬍ 6 ⫺x(⫺1) ⬎ 6(⫺1) x ⬎ ⫺6

3x ⬍ 6

1 1 3x a b ⬍ 6 a b 3 3 x⬍2 ⫺3x ⬎ ⫺27 1 1 ⫺3x a⫺ b ⬍ ⫺27 a⫺ b 3 3 x⬍9

For some inequalities, it requires several operations to ﬁnd their solution sets. In this case, the order in which the operations are performed is the same as that used in solving linear equations.

● EXAMPLE 8 Solution of an Inequality Solve the inequality 2(x ⫺ 4) ⬍

x⫺3 . 3

1.1 Solutions of Linear Equations and Inequalities in One Variable ●

67

Solution x⫺3 3 6(x ⫺ 4) ⬍ x ⫺ 3 6x ⫺ 24 ⬍ x ⫺ 3 5x ⬍ 21 21 x⬍ 5

2(x ⫺ 4) ⬍

Clear fractions. Remove parentheses. Perform additions and subtractions. 1 Multiply by . 5

Check: Now, if we want to check that this solution is a reasonable one, we can substitute the integer values around 21兾5 into the original inequality. Note that x ⫽ 4 satisﬁes the inequality because 23(4) ⫺ 44 ⬍

(4) ⫺ 3 3

23(5) ⫺ 44 ⬍

(5) ⫺ 3 3

but that x ⫽ 5 does not because

Thus x ⬍ 21兾5 is a reasonable solution. ✔ We may also solve inequalities of the form a ⱕ b. This means “a is less than b or a equals b.” The solution of 2x ⱕ 4 is x ⱕ 2, because x ⬍ 2 is the solution of 2x ⬍ 4 and x ⫽ 2 is the solution of 2x ⫽ 4.

● EXAMPLE 9 Solution of an Inequality Solve the inequality 3x ⫺ 2 ⱕ 7 and graph the solution. Solution This inequality states that 3x ⫺ 2 ⫽ 7 or 3x ⫺ 2 ⬍ 7. By solving in the usual manner, we get 3x ⱕ 9, or x ⱕ 3. Then x ⫽ 3 is the solution to 3x ⫺ 2 ⫽ 7 and x ⬍ 3 is the solution to 3x ⫺ 2 ⬍ 7, so the solution set for 3x ⫺ 2 ⱕ 7 is 5x: x ⱕ 36. The graph of the solution set includes the point x ⫽ 3 and all points x ⬍ 3 (see Figure 1.1). Figure 1.1 ●

Checkpoint

x −3

−2

−1

0

1

2

3

4

5

6

7

Solve the following inequalities for y. 3. 3y ⫺ 7 ⱕ 5 ⫺ y 4. 2y ⫹ 6 ⬎ 4y ⫹ 5

5. 4 ⫺ 3y ⱖ 4y ⫹ 5

● EXAMPLE 10 Normal Height for a Given Age For boys between 4 and 16 years of age, height and age are linearly related. That relation can be expressed as H ⫽ 2.31A ⫹ 31.26 where H is height in inches and A is age in years. To account for natural variation among individuals, normal is considered to be any measure falling within ⫾5% of the height

68

● Chapter 1

Linear Equations and Functions

obtained from the equation.* Write as an inequality the range of normal height for a boy who is 9 years old. Solution The boy’s height from the formula is H ⫽ 2.31(9) ⫹ 31.26 ⫽ 52.05 inches. For a 9-yearold boy to be considered of normal height, H would have to be within ⫾5% of 52.05 inches. That is, the boy’s height H is considered normal if H ⱖ 52.05 ⫺ (0.05)(52.05) and H ⱕ 52.05 ⫹ (0.05)(52.05). We can express this range of normal height by the compound inequality 52.05 ⫺ (0.05)(52.05) ⱕ H ⱕ 52.05 ⫹ (0.05)(52.05) or 49.45 ⱕ H ⱕ 54.65

●

Checkpoint Solutions

1. (a) 4(x ⫺ 3) ⫽ 10x ⫺ 12 4x ⫺ 12 ⫽ 10x ⫺ 12 ⫺6x ⫽ 0 x⫽0 Check: 4(0 ⫺ 3) ⱨ 10 ⴢ 0 ⫺ 12 ⫺12 ⫽ ⫺12 ✔

(b)

5(x ⫺ 3) x ⫺x⫽1⫺ 6 9 LCD of 6 and 9 is 18. 5x ⫺ 15 x 18a b ⫺ 18(x) ⫽ 18(1) ⫺ 18a b 6 9 3(5x ⫺ 15) ⫺ 18x ⫽ 18 ⫺ 2x 15x ⫺ 45 ⫺ 18x ⫽ 18 ⫺ 2x ⫺45 ⫺ 3x ⫽ 18 ⫺ 2x ⫺63 ⫽ x 5(⫺63 ⫺ 3) (⫺63) Check: ⫺ (⫺63) ⱨ 1 ⫺ 6 9 5(⫺11) ⫹ 63 ⱨ 1 ⫹ 7 8⫽8✔

x 2x ⫽2⫺ 3x ⫺ 6 x⫺2 LCD is 3x ⫺ 6, or 3(x ⫺ 2). x 2x (3x ⫺ 6)a b ⫽ (3x ⫺ 6)(2) ⫺ (3x ⫺ 6)a b 3x ⫺ 6 x⫺2 x ⫽ 6x ⫺ 12 ⫺ 3(2x) or x ⫽ 6x ⫺ 12 ⫺ 6x x ⫽ ⫺12 2(⫺12) ⫺12 Check: ⱨ2⫺ 3(⫺12) ⫺ 6 ⫺12 ⫺ 2 ⫺12 12 ⱨ2⫺ ⫺42 7 2 2 ⫽ ✔ 7 7 2. y ⫺ 4 ⫽ ⫺4(x ⫹ 2) y ⫺ 4 ⫽ ⫺4x ⫺ 8 y ⫽ ⫺4x ⫺ 4 3. 3y ⫺ 7 ⱕ 5 ⫺ y 4. 2y ⫹ 6 ⬎ 4y ⫹ 5 5. 4 ⫺ 3y ⱖ 4y ⫹ 5 4y ⫺ 7 ⱕ 5 6 ⬎ 2y ⫹ 5 4 ⱖ 7y ⫹ 5 4y ⱕ 12 1 ⬎ 2y ⫺1 ⱖ 7y 1 1 yⱕ3 y⬍ yⱕ⫺ 2 7 (c)

*Adapted from data from the National Center for Health Statistics

1.1 Solutions of Linear Equations and Inequalities in One Variable ●

69

1.1 Exercises In Problems 1–12, solve each equation. 1. 4x ⫺ 7 ⫽ 8x ⫹ 2 2. 3x ⫹ 22 ⫽ 7x ⫹ 2 3. x ⫹ 8 ⫽ 8(x ⫹ 1) 4. x ⫹ x ⫹ x ⫽ x 3 1 5. ⫺ x ⫽ 24 6. ⫺ x ⫽ 12 4 6 7. 2(x ⫺ 7) ⫽ 5(x ⫹ 3) ⫺ x 8. 3(x ⫺ 4) ⫽ 4 ⫺ 2(x ⫹ 2) 5x 2x ⫺ 7 2x x⫺2 9. ⫺4⫽ 10. ⫺1⫽ 2 6 3 2 1 2 11. x ⫹ ⫽ 2 ax ⫺ b ⫺ 6x 3 3 3x 1 2 1 12. ⫺ ⫽ 1 ⫺ ax ⫺ b 4 3 3 6

The equations in Problems 13–18 lead to linear equations. Because not all solutions to the linear equations are solutions to the original equations, be sure to check the solutions in the original equations. 33 ⫺ x 3x ⫹ 3 13. 14. ⫽2 ⫽7 5x x⫺3 1 2x 2 5 3 2 1 15. 16. ⫹ ⫽ ⫹ ⫽ ⫺ x x 2x ⫹ 5 3 4x ⫹ 10 4 3 2x 1 5 2 2x 6 17. 18. ⫹ ⫽ ⫹ ⫽4⫹ x⫺1 3 6 x⫺1 x⫺3 x⫺3 In Problems 19–22, use a calculator to solve each equation. Round your answer to three decimal places. 19. 3.259x ⫺ 8.638 ⫽ ⫺3.8(8.625x ⫹ 4.917) 20. 3.319(14.1x ⫺ 5) ⫽ 9.95 ⫺ 4.6x 21. 0.000316x ⫹ 9.18 ⫽ 2.1(3.1 ⫺ 0.0029x) ⫺ 4.68 22. 3.814x ⫽ 2.916(4.2 ⫺ 0.06x) ⫹ 5.3 In Problems 23–26, solve for y in terms of x. 23. 3x ⫺ 4y ⫽ 15 24. 3x ⫺ 5y ⫽ 25 3x 3 1 25. 9x ⫹ y ⫽ 11 26. ⫹ 5y ⫽ 2 2 3 27. Solve S ⫽ P ⫹ Prt for t. y⫺b ⫽ m for y. 28. Solve x⫺a In Problems 29–34, solve each inequality. 29. 3(x ⫺ 1) ⬍ 2x ⫺ 1 30. 2(x ⫹ 1) ⬎ x ⫺ 1 31. 1 ⫺ 2x ⬎ 9 32. 17 ⫺ x ⬍ ⫺4 3(x ⫺ 1) x⫺1 33. ⱕx⫺2 34. ⫹1⬎x⫹1 2 2 In Problems 35–40, solve each inequality and graph the solution. 35. 2(x ⫺ 1) ⫺ 3 ⬎ 4x ⫹ 1 36. 7x ⫹ 4 ⱕ 2(x ⫺ 1) ⫺3x ⫺2x ⬎3⫺x ⱕ ⫺10 ⫺ x 37. 38. 2 5

39.

2(x ⫺ 1) 3x 1 ⫺ ⬍x⫺ 4 6 3

40.

4x 1 5x ⫺3⬎ ⫹ 3 2 12

A P P L I C AT I O N S 41. Depreciation A $648,000 property is depreciated for tax purposes by its owner with the straight-line depreciation method. The value of the building, y, after x months of use is given by y ⫽ 648,000 ⫺ 1800x dollars. After how many months will the value of the building be $387,000? 42. Depreciation When an $810,000 building is depreciated for tax purposes (by the straight-line method), its value, y, after x months of use is given by y ⫽ 810,000 ⫺ 2250x. How many months will it be before the building is fully depreciated (that is, its value is $0)? How many years is this? 43. Credit card debt High interest rates make it difﬁcult for people to pay off credit card debt in a reasonable period of time. The interest I (in dollars) paid on a $10,000 debt over 3 years when the interest rate is r% can be approximated by the equation I ⫹ 0.663 ⫽ r 175.393 (Source: Consumer Federation of America) If the credit card interest rate is 19.8%, ﬁnd the amount of interest paid during the 3 years. 44. Seawater pressure In seawater, the pressure p is related to the depth d according to 33p ⫺ 18d ⫽ 495 where d is in feet and p is in pounds per square inch. (a) Solve this equation for p in terms of d. (b) The Titanic was discovered at a depth of 12,460 ft. Find the pressure at this depth. 45. Break-even Burnem, Inc. manufactures blank CDs and sells them to a distributor in packs of 500 CDs. Burnem’s total cost and total revenue (in dollars) for x packs of 500 CDs are given by Total cost ⫽ 2x ⫹ 7920

and Total revenue ⫽ 20x

How many packs of 500 CDs must Burnem sell to break even? 46. Break-even Dish Systems manufactures satellite systems and has its monthly proﬁt P in dollars related to the number of satellite systems, x, by 4P ⫽ 81x ⫺ 29,970

70

● Chapter 1

Linear Equations and Functions

Find the number of systems that Dish Systems needs to produce and sell in order to break even. 47. Proﬁt In its second year of operation, a local Internet provider’s proﬁts were $170,500. If this amount was 576% of the company’s ﬁrst-year proﬁts, ﬁnd the ﬁrstyear proﬁts (to the nearest hundred dollars). 48. Sales tax The total price of a new car (including 6% sales tax) is $21,041. How much of this is tax? 49. Computers and the Internet The percent of various types and locations of U.S. households with Internet service, I, can be approximated by the equation 959C ⫺ 1000I ⫽ 456.8 where C is the percent of those households with a computer. (Source: National Telecommunications and Information Administration, U.S. Department of Commerce) According to the equation, what percent of households would need to have a computer before the percent with Internet service reached 75%? 50. Median household income The U.S. median household income for blacks can be described by the equation B ⫽ 1.052W ⫺ 18,691.28 where W is the U.S. median household income for whites. (Source: U.S. Bureau of the Census) When black households have a median income of $50,000, what is the predicted median household income for whites? 51. Course grades To earn an A in a course, a student must get at least a 90 average on four tests and a ﬁnal exam, with the ﬁnal exam weighted twice that of any one test. If the four test scores are 93, 69, 89, and 97, what is the lowest score the student can earn on the ﬁnal exam and still get an A in the course? 52. Course grades Suppose a professor counts the ﬁnal exam as being equal to each of the other tests in her course, and she will also change the lowest test score to match the ﬁnal exam score if the ﬁnal exam score is higher. If a student’s four test scores are 83, 67, 52, and 90, what is the lowest score the student can earn on the ﬁnal exam and still obtain at least an 80 average for the course? 53. Investment mix A retired woman has $120,000 to invest. She has chosen one relatively safe investment fund that has an annual yield of 9% and another, riskier fund that has a 13% annual yield. How much should she invest in each fund if she would like to earn $12,000 per year from her investments? 54. Investment yields One safe investment pays 10% per year, and a more risky investment pays 18% per year.

A woman who has $145,600 to invest would like to have an income of $20,000 per year from her investments. How much should she invest at each rate? 55. Salary increases A woman making $2000 per month has her salary reduced by 10% because of sluggish sales. One year later, after a dramatic improvement in sales, she is given a 20% raise over her reduced salary. Find her salary after the raise. What percent change is this from the $2000 per month? 56. Wildlife management In wildlife management, the capture-mark-recapture technique is used to estimate the populations of ﬁsh or birds in an area or to measure the infestation of insects such as Japanese beetles. Suppose 100 individuals of the species being studied are caught, marked, and released, and one week later 100 more are caught. To estimate the total number of individuals, the following relationship is used: Total marked found in 2nd capture Total number marked ⫽ Total in 2nd capture Total population (a) If in the second capture of 100, it is found that 3 are marked, what is the total population? (b) Suppose that 1000 beetles are captured, marked, and released. Suppose further that in the second capture of 1000, it is found that 63 are marked. What is the population estimate? 57. Proﬁt For a certain product, the revenue function is R(x) ⫽ 40x and the cost function is C(x) ⫽ 20x ⫹ 1600. To obtain a proﬁt, the revenue must be greater than the cost. For what values of x will there be a proﬁt? Graph the solution. 58. Car rental Thrift rents a compact car for $33 per day, and General rents a similar car for $20 per day plus an initial fee of $78. For how many days would it be cheaper to rent from General? Graph the solution. 59. Purchasing Sean can spend at most $900 for a video camera and some video tapes. He plans to buy the camera for $695 and tapes for $5.75 each. Write an inequality that could be used to ﬁnd the number of tapes (x) that he could buy. How many tapes could he buy? 60. Taxes In Sweetwater, Arizona, water bills are taxed on the basis of the amount of the monthly bill in order to encourage conservation. If the bill is more than $0 but less than $60, the tax is 2% of the bill; if the bill is $60 but less than $80, the tax is 4% of the bill; and if the bill is $80 or more, the tax is 6% of the bill. Write the inequalities that represent the amounts of tax owed in these three cases.

1.2 Functions ●

61. Wireless service spending The total amount spent in the United States for wireless communication services, S (in billions of dollars), can be modeled (that is, approximated with some accuracy) by S ⫽ 6.205 ⫹ 11.23t where t is the number of years past 1995. (Source: Cellular Telecommunications and Internet Association) (a) What value of t represents the year 2010? (b) What values of t give S ⬎ 250? (c) In what year does this equation project that spending for wireless communication services will exceed $250 billion? 62. Cigarette use The percent p of high school seniors who smoke cigarettes can be modeled by the equation p ⫽ 75.75 ⫺ 0.74t where t is the number of years past 1975. (Source: monitoringthefuture.org) (a) What value of t represents 2010? (b) What percent does this equation predict for 2010? (c) This equation ceases to be effective when p becomes negative and perhaps before that. Find the years when p ⬍ 0. 63. Heat index During the summer of 1998, Dallas, Texas, endured 29 consecutive days on which the temperature

1.2

71

was at least 110°F. On many of those days, the combination of heat and humidity made it feel even hotter than it was. When the temperature is 100°F, the apparent temperature A (or heat index) depends on the humidity h (expressed as a decimal) according to A ⫽ 90.2 ⫹ 41.3h* (a) For what humidity levels is the apparent temperature at least 110°F? (Note that this answer will be a closed interval. Why?) (b) For what humidity levels is the apparent temperature less than 100°F? 64. Wind chill The combination of cold temperatures and wind speed determine what is called wind chill. The wind chill is a temperature that is the still-air equivalent of the combination of cold and wind. When the wind speed is 25 mph, the wind chill WC depends on the temperature t (in degrees Fahrenheit) according to WC ⫽ 1.337t ⫺ 24.094 For what temperatures does it feel at least 30°F colder than the air temperature? That is, ﬁnd t such that WC ⱕ t ⫺ 30. *Source: Bosch, W., and C. G. Cobb, “Temperature-Humidity Indices,” UMAP Unit 691, The UMAP Journal, 10(3), Fall 1989, 237–256.

Functions

OBJECTIVES ●

●

● ●

●

To determine whether a relation is a function To state the domains and ranges of certain functions To use function notation To perform operations with functions To find the composite of two functions

Relations and Functions

] Application Preview The number of individual income tax returns filed electronically has increased in recent years. The relationship between the number of returns filed electronically and the number of years after 1995 can be described by the equation y ⴝ 5.091x ⴙ 11.545 where y is in millions and x is the number of years past 1995. (Source: Internal Revenue Service) In this equation, y depends uniquely on x, so we say that y is a function of x. Understanding the mathematical meaning of the phrase function of and learning to interpret and apply such relationships are the goals of this section. (The function above will be discussed in Example 4.)

An equation or inequality containing two variables expresses a relation between those two variables. For example, the inequality R ⱖ 35x expresses a relation between the two variables x and R, and the equation y ⫽ 4x ⫺ 3 expresses a relation between the two variables x and y.

72

● Chapter 1

Linear Equations and Functions

In addition to deﬁning a relation by an equation, an inequality, or rule of correspondence, we may also deﬁne it as any set of ordered pairs of real numbers (a, b). For example, the solutions to y ⫽ 4x ⫺ 3 are pairs of numbers (one for x and one for y). We write the pairs (x, y) so that the ﬁrst number is the x-value and the second is the y-value, and these ordered pairs deﬁne the relation between x and y. Some relations may be deﬁned by a table, a graph, or an equation. Relation

A relation is deﬁned by a set of ordered pairs or by a rule that determines how the ordered pairs are found. It may also be deﬁned by a table, a graph, an equation, or an inequality. 5(1, 3), (1, 6), (2, 6), (3, 9), (3, 12), (4, 12)6

For example, the set of ordered pairs

expresses a relation between the set of ﬁrst components, 51, 2, 3, 46 , and the set of second components, 53, 6, 9, 126 . The set of ﬁrst components is called the domain of the relation, and the set of second components is called the range of the relation. Figure 1.2(a) uses arrows to indicate how the inputs from the domain (the ﬁrst components) are associated with the outputs in the range (the second components). Figure 1.2(b) shows another example of a relation. Because relations can also be deﬁned by tables and graphs, Table 1.1 and Figure 1.3 are examples of relations.

Figure 1.2

Domain

Range

Domain

1

3

1

2

6

2

3

9

3

4

12

4

(a)

Range 7 9 11

(b)

An equation frequently expresses how the second component (the output) is obtained from the ﬁrst component (the input). For example, the equation y ⫽ 4x ⫺ 3 expresses how the output y results from the input x. This equation expresses a special relation between x and y, because each value of x that is substituted into the equation results in exactly one value of y. If each value of x put into an equation results in one value of y, we say that the equation expresses y as a function of x. Definition of a Function

A function is a relation between two sets such that to each element of the domain (input) there corresponds exactly one element of the range (output). A function may be deﬁned by a set of ordered pairs, a table, a graph, or an equation. When a function is deﬁned, the variable that represents the numbers in the domain (input) is called the independent variable of the function, and the variable that represents the numbers in the range (output) is called the dependent variable (because its values depend on the values of the independent variable). The equation y ⫽ 4x ⫺ 3 deﬁnes y as a function of x, because only one value of y will result from each value of x that is substituted

1.2 Functions ●

73

TABLE 1.1

2007 Federal Income Tax Income

Rate

$0–$7,825 $7,826–$31,850 $31,851–$77,100 $77,101–$160,850 $160,851–$349,700 $349,700⫹

10% 15% 25% 28% 33% 35%

Source: Internal Revenue Service

Figure 1.3 Source: The Wall Street Journal, January 17, 2002. Copyright © 2002 Dow Jones & Co. Reprinted by permission of Dow Jones & Co. via Copyright Clearance Center.

into the equation. Thus the equation deﬁnes a function in which x is the independent variable and y is the dependent variable. We can also apply this idea to a relation deﬁned by a table or a graph. In Figure 1.2(b), because each input in the domain corresponds to exactly one output in the range, the relation is a function. Similarly, the data given in Table 1.1 (the tax brackets for U.S. income tax for single wage earners) represents the tax rate as a function of the income. Note in Table 1.1 that even though many different amounts of taxable income have the same tax rate, each amount of taxable income (input) corresponds to exactly one tax rate (output). Thus tax rate (the dependent variable) is a function of a single ﬁler’s taxable income (the independent variable). On the other hand, the relation deﬁned in Figure 1.3 is not a function because the graph representing the Dow Jones Utilities Average shows that for each day there are at least three different values—the actual high, the actual low, and the close. For example, on September 25, 2001, the average varies from a low of 286 to a high of 298. Graphs such as Figure 1.3 appear in many daily newspapers, so current versions are readily available. However, this particular ﬁgure also has historical interest because it shows a break in the graph when the New York Stock Exchange closed following the terrorist attacks of 9/11/2001.

● EXAMPLE 1 Functions Does y2 ⫽ 2x express y as a function of x? Solution No, because some values of x are associated with more than one value of y. In fact, there are two y-values for each x ⬎ 0. For example, if x ⫽ 8, then y ⫽ 4 or y ⫽ ⫺4, two different y-values for the same x-value. The equation y2 ⫽ 2x expresses a relation between x and y, but y is not a function of x.

Graphs of Functions

It is possible to picture geometrically the relations and functions that we have been discussing by sketching their graphs on a rectangular coordinate system. We construct a rectangular coordinate system by drawing two real number lines (called coordinate axes) that are perpendicular to each other and intersect at their origins (called the origin of the system).

74

● Chapter 1

Linear Equations and Functions

The ordered pair (a, b) represents the point P that is located a units along the x-axis and b units along the y-axis (see Figure 1.4). Similarly, any point has a unique ordered pair that describes it. y-axis Quadrant II 5

Quadrant I

4 3

A(5, 2)

2

B(−4, 1)

1 −5 −4 −3 −2 −1 −1

x-axis 1

2

3

4

5

−2 −3

C(−4, −4)

Figure 1.4

−4 −5

Quadrant III

D(1, −5) Quadrant IV

The values a and b in the ordered pair associated with the point P are called the rectangular (or Cartesian) coordinates of the point, where a is the x-coordinate (or abscissa), and b is the y-coordinate (or ordinate). The ordered pairs (a, b) and (c, d) are equal if and only if a ⫽ c and b ⫽ d. The graph of an equation that deﬁnes a function (or relation) is the picture that results when we plot the points whose coordinates (x, y) satisfy the equation. To sketch the graph, we plot enough points to suggest the shape of the graph and draw a smooth curve through the points. This is called the point-plotting method of sketching a graph.

● EXAMPLE 2 Graphing a Function Graph the function y ⫽ 4x2 . Solution We choose some sample values of x and ﬁnd the corresponding values of y. Placing these in a table, we have sample points to plot. When we have enough to determine the shape of the graph, we connect the points to complete the graph. The table and graph are shown in Figure 1.5(a). y

y

x –1 – 12 0 1 2

1 4 Figure 1.5

y 4 1 0 1 4 16

14

4

12

3

10

2

8

1

y 2 = 2x x

6 −1

4

y = 4x 2

2

1

3

−2 −3

−4 −3 −2 −1 −2

(a)

1

2

3

4

−4

(b)

4

5

6

7

1.2 Functions ●

75

We can determine whether a relation is a function by inspecting its graph. If the relation is a function, then no one input (x-value) has two different outputs (y-values). This means that no two points on the graph will have the same ﬁrst coordinate (component). Thus no two points of the graph will lie on the same vertical line. Vertical-Line Test

If no vertical line exists that intersects the graph at more than one point, then the graph is that of a function. Performing this test on the graph of y ⫽ 4x2 (Figure 1.5(a)), we easily see that this equation describes a function. The graph of y2 ⫽ 2x is shown in Figure 1.5(b), and we can see that the vertical-line test indicates that this is not a function (as we already saw in Example 1). For example, a vertical line at x ⫽ 2 intersects the curve at (2, 2) and (2, ⫺2). The graph in Figure 1.6 shows one possible projection of the resources of the Pension Beneﬁt Guaranty Corporation, the agency that insures pensions, (y) versus time (x). It represents a function, because for every x there is exactly one y. On the other hand, as noted previously, the graph in Figure 1.3 (earlier in this section) does not represent a function.

Function Notation

+$50 billion

We can use function notation to indicate that y is a function of x. The function is denoted by f, and we write y ⫽ f(x). This is read “y is a function of x” or “y equals f of x.” For speciﬁc values of x, f(x) represents the values of the function (that is, outputs, or y-values) at those x-values. Thus if f(x) ⫽ 3x2 ⫹ 2x ⫹ 1 f(2) ⫽ 3(2)2 ⫹ 2(2) ⫹ 1 ⫽ 17 f(⫺3) ⫽ 3(⫺3)2 ⫹ 2(⫺3) ⫹ 1 ⫽ 22

then and

+ 40 + 30

Figure 1.7 shows the function notation f(x) as (a) an operator on x and (b) a y-coordinate for a given x-value.

+ 20 + 10

y

0 – 10 ’04

’08

’12

’16

’20

(−3, f(−3))

input x

y = f(x) 3 2

Figure 1.6 Source: New York Times, September 14, 2004. Copyright © 2004 The New York Times Co. Reprinted by permission.

1

(0, f(0))

f −3 −2 −1

2

x −1

output f(x) (a)

1

3

(2, f(2))

(b)

Figure 1.7

Letters other than f may also be used to denote functions. For example, y ⫽ g(x) or y ⫽ h(x) may be used.

● EXAMPLE 3 Evaluating Functions If y ⫽ f(x) ⫽ 2x3 ⫺ 3x2 ⫹ 1, ﬁnd the following. (a) f(3) (b) f(⫺1) (c) f(a) (d) f(⫺a) Solution (a) f(3) ⫽ 2(3)3 ⫺ 3(3)2 ⫹ 1 ⫽ 2(27) ⫺ 3(9) ⫹ 1 ⫽ 28 Thus y ⫽ 28 when x ⫽ 3.

76

● Chapter 1

Linear Equations and Functions

(b) f(⫺1) ⫽ 2(⫺1)3 ⫺ 3(⫺1)2 ⫹ 1 ⫽ 2(⫺1) ⫺ 3(1) ⫹ 1 ⫽ ⫺4 Thus y ⫽ ⫺4 when x ⫽ ⫺1. (c) f(a) ⫽ 2a3 ⫺ 3a2 ⫹ 1 (d) f(⫺a) ⫽ 2(⫺a)3 ⫺ 3(⫺a)2 ⫹ 1 ⫽ ⫺2a3 ⫺ 3a2 ⫹ 1 ] EXAMPLE 4 Electronic Income Tax Returns (Application Preview) The relationship between the number of individual income tax returns ﬁled electronically and the number of years after 1995 can be described by the equation y ⫽ f(x) ⫽ 5.091x ⫹ 11.545 where y is in millions and x is the number of years after 1995. (Source: Internal Revenue Service) (a) Find f(8). (b) Write a sentence that explains the meaning of the result in (a). Solution (a) f(8) ⫽ 5.091(8) ⫹ 11.545 ⫽ 52.273 (b) The statement f(8) ⫽ 52.273 means that in 1995 ⫹ 8 ⫽ 2003, 52.273 million income tax returns were ﬁled electronically.

● EXAMPLE 5 Mortgage Payment TABLE 1.2 r(%)

f(r)

2.6 5.2 6.3 7.4 9

12 15 17 20 30

Table 1.2 shows the number of years that it will take a couple to pay off a $100,000 mortgage at several different interest rates if they pay $800 per month. If r denotes the rate and f(r) denotes the number of years: (a) What is f(6.3) and what does it mean? (b) If f(r) ⫽ 30, what is r? (c) Does 2 ⴢ f(2.6) ⫽ f(2 ⴢ 2.6)? Solution (a) f(6.3) ⫽ 17. This means that with a 6.3% interest rate, a couple can pay off the $100,000 mortgage in 17 years by paying $800 per month. (b) The table indicates that f(9) ⫽ 30, so r ⫽ 9. (c) 2 ⴢ f(2.6) ⫽ 2 ⴢ 12 ⫽ 24 and f(2 ⴢ 2.6) ⫽ f(5.2) ⫽ 15, so 2 ⴢ f(2.6) ⫽ f(2 ⴢ 2.6).

● EXAMPLE 6 Function Notation Given f(x) ⫽ x2 ⫺ 3x ⫹ 8, ﬁnd

f(x ⫹ h) ⫺ f(x) and simplify (if h ⫽ 0). h

Solution We ﬁnd f(x ⫹ h) by replacing each x in f(x) with the expression x ⫹ h.

3(x ⫹ h)2 ⫺ 3(x ⫹ h) ⫹ 84 ⫺ 3x2 ⫺ 3x ⫹ 84 f(x ⫹ h) ⫺ f(x) ⫽ h h ⫽

3(x2 ⫹ 2xh ⫹ h2) ⫺ 3x ⫺ 3h ⫹ 84 ⫺ x2 ⫹ 3x ⫺ 8 h

x2 ⫹ 2xh ⫹ h2 ⫺ 3x ⫺ 3h ⫹ 8 ⫺ x2 ⫹ 3x ⫺ 8 h 2 h(2x ⫹ h ⫺ 3) 2xh ⫹ h ⫺ 3h ⫽ ⫽ ⫽ 2x ⫹ h ⫺ 3 h h ⫽

1.2 Functions ●

Domains and Ranges

77

We will limit our discussion in this text to real functions, which are functions whose domains and ranges contain only real numbers. If the domain and range of a function are not speciﬁed, it is assumed that the domain consists of all real inputs (x-values) that result in real outputs (y-values), making the range a subset of the real numbers. For the types of functions we are now studying, if the domain is unspeciﬁed, it will include all real numbers except 1. values that result in a denominator of 0, and 2. values that result in an even root of a negative number.

● EXAMPLE 7 Domain and Range Find the domain of each of the following functions; ﬁnd the range for the functions in (a) and (b). (a) y ⫽ 4x2

(b) y ⫽ 14 ⫺ x

(c) y ⫽ 1 ⫹

1 x⫺2

Solution (a) There are no restrictions on the numbers substituted for x, so the domain consists of all real numbers. Because the square of any real number is nonnegative, 4x2 must be nonnegative. Thus the range is y ⱖ 0. If we plot points or use a graphing utility, we will get the graph shown in Figure 1.8(a), which illustrates our conclusions about the domain and range. (b) We note the restriction that 4 ⫺ x cannot be negative. Thus the domain consists of only numbers less than or equal to 4. That is, the domain is the set of real numbers satisfying x ⱕ 4. Because 14 ⫺ x is always nonnegative, the range is all y ⱖ 0. Figure 1.8(b) shows the graph of y ⫽ 14 ⫺ x. Note that the graph is located only where x ⱕ 4 and on or above the x-axis (where y ⱖ 0). 1 1 (c) y ⫽ 1 ⫹ is undeﬁned at x ⫽ 2 because is undeﬁned. Hence, the domain x⫺2 0 consists of all real numbers except 2. Figure 1.8(c) shows the graph of 1 y⫽1⫹ . The break where x ⫽ 2 indicates that x ⫽ 2 is not part of the domain. x⫺2 y

y

14

8

y = 1+

12 10

4

8

2

6

x −8 −6 −4 −2

2

−2

4

y = 4x 2

2

−4 x

−4 −3 −2 −1

1

−2

(a) Figure 1.8

1 6 x−2

2

3

4

−6 −8

(c)

4

6

8

78

● Chapter 1 ●

Linear Equations and Functions

Checkpoint

Operations with Functions

Operations with Functions

1. 2. 3. 4.

If If If If

y ⫽ f(x), the independent variable is _______ and the dependent variable is _______. (1, 3) is on the graph of y ⫽ f(x), then f(1) ⫽ ? f(x) ⫽ 1 ⫺ x3, ﬁnd f(⫺2). f(x) ⫽ 2x2, ﬁnd f(x ⫹ h). 1 5. If f(x) ⫽ , what is the domain of f(x)? x⫹1 We can form new functions by performing algebraic operations with two or more functions. We deﬁne new functions that are the sum, difference, product, and quotient of two functions as follows. Let f and g be functions of x, and deﬁne the following. Sum Difference Product Quotient

( f ⫹ g)(x) ⫽ f(x) ⫹ g(x) ( f ⫺ g)(x) ⫽ f(x) ⫺ g(x) ( f ⴢ g)(x) ⫽ f(x) ⴢ g(x) f f(x) a b(x) ⫽ g g(x)

if g(x) ⫽ 0

● EXAMPLE 8 Operations with Functions If f(x) ⫽ 3x ⫹ 2 and g(x) ⫽ x2 ⫺ 3, ﬁnd the following functions. (a) ( f ⫹ g)(x)

(b) ( f ⫺ g)(x)

(c) ( f ⴢ g)(x)

f (d) a b(x) g

Solution (a) ( f ⫹ g)(x) ⫽ f(x) ⫹ g(x) ⫽ (3x ⫹ 2) ⫹ (x2 ⫺ 3) ⫽ x2 ⫹ 3x ⫺ 1 (b) ( f ⫺ g)(x) ⫽ f(x) ⫺ g(x) ⫽ (3x ⫹ 2) ⫺ (x2 ⫺ 3) ⫽ ⫺x2 ⫹ 3x ⫹ 5 (c) ( f ⴢ g)(x) ⫽ f(x) ⴢ g(x) ⫽ (3x ⫹ 2)(x2 ⫺ 3) ⫽ 3x3 ⫹ 2x2 ⫺ 9x ⫺ 6 f(x) f 3x ⫹ 2 (d) a b(x) ⫽ ⫽ 2 , if x2 ⫺ 3 ⫽ 0 g g(x) x ⫺3 We now consider a new way to combine two functions. Just as we can substitute a number for the independent variable in a function, we can substitute a second function for the variable. This creates a new function called a composite function. Composite Functions

Let f and g be functions. Then the composite functions g of f (denoted g ⴰ f ) and f of g (denoted f ⴰ g) are deﬁned as follows: (g ⴰ f )(x) ⫽ g( f(x)) ( f ⴰ g)(x) ⫽ f (g(x)) Note that the domain of g ⴰ f is the subset of the domain of f for which g ⴰ f is deﬁned. Similarly, the domain of f ⴰ g is the subset of the domain of g for which f ⴰ g is deﬁned.

1.2 Functions ●

79

● EXAMPLE 9 Composite Functions If f(x) ⫽ 2x3 ⫹ 1 and g(x) ⫽ x2, ﬁnd the following. (a) (g ⴰ f )(x) (b) ( f ⴰ g)(x) Solution (a) (g ⴰ f )(x) ⫽ g( f(x)) ⫽ g(2x3 ⫹ 1) ⫽ (2x3 ⫹ 1)2 ⫽ 4x6 ⫹ 4x3 ⫹ 1

(b) ( f ⴰ g)(x) ⫽ f(g(x)) ⫽ f(x2) ⫽ 2(x2)3 ⫹ 1 ⫽ 2x6 ⫹ 1

Figure 1.9 illustrates both composite functions found in Example 9. (g

f )(x) x

g

f(x) = 2x 3 + 1

g

●

●

Checkpoint

Checkpoint Solutions

g)(x) x

f

Figure 1.9

(f

g( f(x)) = (2x 3 + 1)2 (g f )(x) = 4x 6 + 4x 3 + 1

g(x) = x 2

f

f(g(x)) = 2(x 2)3 + 1 ( f g)(x) = 2x 6 + 1

6. If f(x) ⫽ 1 ⫺ 2x and g(x) ⫽ 3x2, ﬁnd the following. (a) (g ⫺ f )(x) (b) ( f ⴢ g)(x) (c) ( f ⴰ g)(x) (d) (g ⴰ f )(x) 1. 2. 3. 4.

(e) ( f ⴰ f )(x) ⫽ f( f(x))

Independent variable is x; dependent variable is y. f(1) ⫽ 3 f (⫺2) ⫽ 1 ⫺ (⫺2)3 ⫽ 1 ⫺ (⫺8) ⫽ 9 f(x ⫹ h) ⫽ 2(x ⫹ h)2 ⫽ 2(x2 ⫹ 2xh ⫹ h2) 5. The domain is all real numbers except x ⫽ ⫺1, because f(x) is undeﬁned when x ⫽ ⫺1. 6. (a) (g ⫺ f )(x) ⫽ g(x) ⫺ f(x) ⫽ 3x2 ⫺ (1 ⫺ 2x) ⫽ 3x2 ⫹ 2x ⫺ 1 (b) ( f ⴢ g)(x) ⫽ f(x) ⴢ g(x) ⫽ (1 ⫺ 2x)(3x2) ⫽ 3x2 ⫺ 6x3 (c) ( f ⴰ g)(x) ⫽ f(g(x)) ⫽ f(3x2) ⫽ 1 ⫺ 2(3x2) ⫽ 1 ⫺ 6x2 (d) (g ⴰ f )(x) ⫽ g( f(x)) ⫽ g(1 ⫺ 2x) ⫽ 3(1 ⫺ 2x)2 (e) ( f ⴰ f )(x) ⫽ f( f(x)) ⫽ f(1 ⫺ 2x) ⫽ 1 ⫺ 2(1 ⫺ 2x) ⫽ 4x ⫺ 1

80

● Chapter 1

Linear Equations and Functions

1.2 Exercises In Problems 1 and 2, use the values in the following table. x

⫺7

⫺1

0

3

4.2

9

11

14

18

22

y

0

0

1

5

9

11

35

22

22

60

1. (a) (b) (c) 2. (a)

Explain why the table deﬁnes y as a function of x. State the domain and range of this function. If the table expresses y ⫽ f(x), ﬁnd f(0) and f(11). If the function deﬁned by the table is denoted by f, so that y ⫽ f(x), is f(9) an input or an output of f ? (b) Does the table describe x as a function of y? Explain.

In Problems 3 and 4, are the relations deﬁned by the tables functions? Explain why or why not and give the domain and range. 3. x

1

2

3

8

9

y

⫺4

⫺4

5

16

5

4. x

⫺1

0

1

3

1

y

0

2

4

6

9

7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

5. Does either of the graphs in Figure 1.10 represent y as a function of x? Explain your choices. y

(9, 10)

12 10 8 6 4 2

4

(5, 6)

2

2 x

-6

18.

y

2 4 6 8 10 12 -4 -6

x -2

6 -2

(2, 2)

19.

(a)

(b)

Figure 1.10

20.

6. Does either of the graphs in Figure 1.11 represent y as a function of x? Explain your choices. y

y

21.

10 8 6 4 2

4 2

22. x

x -4

2 -2 -4

(a) Figure 1.11

4

-8 -4 -4 -6 -8 -10

2

(b)

6 8 10

If y ⫽ 3x3, is y a function of x? If y ⫽ 6x2, is y a function of x? If y2 ⫽ 3x, is y a function of x? If y2 ⫽ 10x2, is y a function of x? If R(x) ⫽ 8x ⫺ 10, ﬁnd the following. (a) R(0) (b) R(2) (c) R(⫺3) (d) R(1.6) If h(x) ⫽ 3x2 ⫺ 2x, ﬁnd the following. (a) h(3) (b) h(⫺3) (c) h(2) (d) h(16) If C(x) ⫽ 4x2 ⫺ 3, ﬁnd the following. (a) C(0) (b) C(⫺1) (c) C(⫺2) (d) C (⫺23) If R(x) ⫽ 100x ⫺ x3, ﬁnd the following. (a) R(1) (b) R(10) (c) R(2) (d) R(⫺10) If f(x) ⫽ x3 ⫺ 4兾x, ﬁnd the following. (a) f (⫺12) (b) f(2) (c) f (⫺2) If C(x) ⫽ (x2 ⫺ 1)兾x, ﬁnd the following. (a) C(1) (b) C (12) (c) C(⫺2) Let f(x) ⫽ 1 ⫹ x ⫹ x2 and h ⫽ 0. (a) Is f(2 ⫹ 1) ⫽ f(2) ⫹ f(1)? (b) Find f(x ⫹ h). (c) Does f(x ⫹ h) ⫽ f(x) ⫹ f(h)? (d) Does f(x ⫹ h) ⫽ f(x) ⫹ h? f(x ⫹ h) ⫺ f(x) (e) Find and simplify. h 2 Let f(x) ⫽ 3x ⫺ 6x and h ⫽ 0. (a) Is f(3 ⫹ 2) ⫽ f(3) ⫹ 2? (b) Find f(x ⫹ h). (c) Does f(x ⫹ h) ⫽ f(x) ⫹ h? (d) Does f(x ⫹ h) ⫽ f(x) ⫹ f(h)? f(x ⫹ h) ⫺ f(x) (e) Find and simplify. h If f(x) ⫽ x ⫺ 2x2 and h ⫽ 0, ﬁnd the following and simplify. f(x ⫹ h) ⫺ f(x) (a) f(x ⫹ h) (b) h If f(x) ⫽ 2x2 ⫺ x ⫹ 3 and h ⫽ 0, ﬁnd the following and simplify. f(x ⫹ h) ⫺ f(x) (a) f(x ⫹ h) (b) h If y ⫽ f (x) in Figure 1.10(a), ﬁnd the following. (a) f(9) (b) f(5) Suppose y ⫽ g(x) in Figure 1.11(b). (a) Find g(0). (b) How many x-values in the domain of this function satisfy g(x) ⫽ 0?

1.2 Functions ●

23. The graph of y ⫽ x2 ⫺ 4x is shown in Figure 1.12. y

y = x 2 – 4x

In Problems 29–32, a function and its graph are given. In each problem, ﬁnd the domain. 1x ⫺ 1 x⫹1 29. f(x) ⫽ 30. f(x) ⫽ x⫺2 1x ⫹ 3

P(a, b) y

y

x -1

2

3

4

5

10

10

5

5

-2

Q

81

x

R

4

6

8

10 12

x -4

-5

-4

2

4

6

8 10

–5

-10

Figure 1.12

(a) What are the coordinates of the point Q? Do they satisfy the equation? (b) What are the coordinates of R? Do they satisfy the equation? (c) If the coordinates of the point P on the graph are (a, b), how are a and b related? (d) What are the x-values of the points on the graph whose y-coordinates are 0? Are these x-values solutions to the equation x2 ⫺ 4x ⫽ 0? 24. The graph of y ⫽ 2x2 is shown in Figure 1.13.

31. f(x) ⫽ 4 ⫹ 149 ⫺ x2

32. f(x) ⫽ ⫺2 ⫺ 19 ⫺ x2 y

y

x

12

-4

-4

1 2 3 4

8

-2

4

-4 x

-8

-2

-6

2 4 6 8

y 6

P(a, b)

5 4

y = 2x 2

3 2

R x

-3 -2 -1

1

2

3

Figure 1.13

(a) Does the point (1, 1) lie on the graph? Do the coordinates satisfy the equation? (b) What are the coordinates of point R? Do they satisfy the equation? (c) If the point P, with coordinates (a, b), is on the graph, how are a and b related? (d) What is the x-value of the point whose y-coordinate is 0? Does this value of x satisfy the equation 0 ⫽ 2x2? State the domain and range of each of the functions in Problems 25–28. 25. y ⫽ x2 ⫹ 4 26. y ⫽ x2 ⫺ 1 27. y ⫽ 1x ⫹ 4 28. y ⫽ 1x2 ⫹ 1

For f(x) and g(x) given in Problems 33–36, ﬁnd (a) ( f ⴙ g)(x) (b) ( f ⴚ g)(x) (c) ( f ⴢ g)(x) (d) ( f兾g)(x) g(x) ⫽ x3 33. f(x) ⫽ 3x g(x) ⫽ 1兾x 34. f(x) ⫽ 1x g(x) ⫽ x2 35. f(x) ⫽ 12x 2 g(x) ⫽ 1 ⫺ 2x 36. f(x) ⫽ (x ⫺ 1) For f(x) and g(x) given in Problems 37–40, ﬁnd (a) ( f ⴰ g)(x) (b) (g ⴰ f )(x) (c) f( f(x)) (d) f 2(x) ⴝ ( f ⴢ f )(x) g(x) ⫽ 1 ⫺ 2x 37. f(x) ⫽ (x ⫺ 1)3 g(x) ⫽ x3 ⫺ 1 38. f(x) ⫽ 3x g(x) ⫽ x4 ⫹ 5 39. f(x) ⫽ 2 1x 1 g(x) ⫽ 4x ⫹ 1 40. f (x) ⫽ 3 x A P P L I C AT I O N S 41. Mortgage A couple seeking to buy a home decides that a monthly payment of $800 ﬁts their budget. Their bank’s interest rate is 7.5%. The amount they can borrow, A, is a function of the time t, in years, it will take

● Chapter 1

Linear Equations and Functions

to repay the debt. If we denote this function by A ⫽ f(t), then the following table deﬁnes the function. t

A

t

A

5 10 15

40,000 69,000 89,000

20 25 30

103,000 113,000 120,000

Source: Comprehensive Mortgage Payment Tables, Publication No. 492, Financial Publishing Co., Boston

(a) Find f (20) and write a sentence that explains its meaning. (b) Does f(5 ⫹ 5) ⫽ f(5) ⫹ f(5)? Explain. (c) If the couple is looking at a house that requires them to ﬁnance $89,000, how long must they make payments? Write this correspondence in the form A ⫽ f(t). 42. Debt reﬁnancing When a debt is reﬁnanced, sometimes the term of the loan (that is, the time it takes to repay the debt) is shortened. Suppose the current interest rate is 7%, and the current debt is $100,000. The monthly payment R of the reﬁnanced debt is a function of the term of the loan, t, in years. If we represent this function by R ⫽ f(t), then the following table deﬁnes the function. t

R

t

R

5 10 12

1980.12 1161.09 1028.39

15 20 25

898.83 775.30 706.78

Source: Comprehensive Mortgage Payment Tables, Publication No. 492, Financial Publishing Co., Boston

were graphed, what parts of the new graph must be the same as this graph and what parts might be the same? Explain. (d) Find the domain and range of n ⫽ f(t) if the function is deﬁned by the graph. 18

Workers per beneficiary

82

16.5

16 14 12 10 8 6

3.4

1.9

4 2

0 1950 1970 1990 2010 2030 2050

Year Source: Social Security Administration

44. Dow Jones Industrial Average If t represents the number of hours after 9:30 A.M. on Wednesday, October 3, 2007, then the graph deﬁnes the Dow Jones Industrial Average D as a function of time t. If we represent this function by D ⫽ f(t), use the graph to complete the following. (a) Find f(0) and f(6.5). (b) Find the domain and range for D ⫽ f(t) as deﬁned by the graph. (c) About how many t-values satisfy f (t) ⫽ 14,000? Estimate one such t-value. THE DOW MINUTE-BY-MINUTE

(a) If they reﬁnance for 20 years, what is the monthly payment? Write this correspondence in the form R ⫽ f(t). (b) Find f (10) and write a sentence that explains its meaning. (c) Is f(5 ⫹ 5) ⫽ f(5) ⫹ f(5)? Explain. 43. Social Security beneﬁts funding Social Security beneﬁts paid to eligible beneﬁciaries are funded by individuals who are currently employed. The following graph, based on known data until 2005, with projections into the future, deﬁnes a function that gives the number of workers, n, supporting each retiree as a function of time t (given by calendar year). Let us denote this function by n ⫽ f (t). (a) Find f(1950) and explain its meaning. (b) Find f (1990). (c) If, after the year 2050, actual data through 2050 regarding workers per Social Security beneﬁciary

Position of the Dow Jones industrial average at 1-minute intervals yesterday. 14,100 14,050

Previous close: 14,047.31

14,000

13,950

79.26

13,900

10 a.m.

Noon

2 p.m.

4 p.m.

Sources: Associated Press; Bloomberg Financial Markets, New York Times, October 4, 2007. Copyright © 2007 The New York Times Co. Reprinted by permission.

45. Imprisonment and parole The ﬁgure on the next page shows the number of persons in state prisons at year’s end and the number of parolees at year’s end, both as functions of the years past 1900. If y ⫽ f(t) gives the

1.2 Functions ●

number of prisoners and y ⫽ g(t) gives the number of parolees, use the ﬁgure to complete the following. (a) Estimate f(95) and g(95). (b) Find f(100) and explain its meaning. (c) Find g(90) and explain its meaning. (d) Find ( f ⫺ g)(100) and explain its meaning. (e) Which of ( f ⫺ g)(93) and ( f ⫺ g)(98) is greater? Explain. Imprisonment outpaces release on parole Number in state prisons and on parole annually at year's end:

1.2

in prison on parole

(in millions)

0.7

(a) Based on the formula for f(s) and the physical context of the problem, what is the domain of f(s)? (b) Find f (10) and write a sentence that explains its meaning. (c) The working domain of this wind chill function is actually s ⱖ 4. How can you tell that s ⫽ 0 is not in the working domain, even though it is in the mathematical domain?

gives the relation between temperature readings in Celsius and Fahrenheit. (a) Is C a function of F? (b) What is the domain? (c) If we consider this equation as relating temperatures of water in its liquid state, what are the domain and range? (d) What is C when F ⫽ 40°?

0 ’90 ’91 ’92 ’93 ’94 ’95 ’96 ’97 ’98 ’99 ’00 Source: Imprisonment outpaces release on parole from USA Today, January 9, 2000. Copyright © 2000. Reprinted by permission of USA Today.

46. Women in the work force The number (in millions) of women in the work force, given as a function f of the year for selected years from 1920 to 2000, is shown in the ﬁgure below. (a) How many women were in the labor force in 1970? (b) Estimate f(1930) and write a sentence that explains its meaning. (c) Estimate f(1990) ⫺ f(1980) and explain its meaning. Number of women in the work force (millions)

WC ⫽ f(s) ⫽ 45.694 ⫹ 1.75s ⫺ 29.261s

5 160 C⫽ F⫺ 9 9

0.7

0.5

⫺5°F, then the wind chill, WC, is a function of the wind speed, s (in mph), and is given by

48. Temperature measurement The equation 1.2

0.8

0.4

83

70 60 50 40 30 20 10 0

49. Cost The total cost of producing a product is given by C(x) ⫽ 300x ⫹ 0.1x2 ⫹ 1200 where x represents the number of units produced. Give (a) the total cost of producing 10 units (b) the value of C(100) (c) the meaning of C(100) 50. Proﬁt The proﬁt from the production and sale of a product is P(x) ⫽ 47x ⫺ 0.01x2 ⫺ 8000, where x represents the number of units produced and sold. Give (a) the proﬁt from the production and sale of 2000 units (b) the value of P(5000) (c) the meaning of P(5000) 51. Pollution Suppose that the cost C (in dollars) of removing p percent of the particulate pollution from the smokestacks of an industrial plant is given by

1920

1940

1960 Year

1980

2000

Source: 2004 World Almanac

47. Wind chill Dr. Paul Siple conducted studies testing the effect of wind on the formation of ice at various temperatures and developed the concept of the wind chill, which we hear reported during winter weather reports. Using Siple’s work, if the air temperature is

C(p) ⫽

7300p 100 ⫺ p

(a) Find the domain of this function. Recall that p represents the percent pollution that is removed. In parts (b)–(e), ﬁnd the functional values and explain what each means. (b) C(45) (c) C(90) (d) C(99) (e) C(99.6)

84

● Chapter 1

Linear Equations and Functions

52. Test reliability If a test that has reliability r is lengthened by a factor n (n ⱖ 1), the reliability R of the new test is given by R(n) ⫽

nr 1 ⫹ (n ⫺ 1)r

0⬍rⱕ1

If the reliability is r ⫽ 0.6, the equation becomes R(n) ⫽

0.6n 0.4 ⫹ 0.6n

(a) Find R(1). (b) Find R(2); that is, ﬁnd R when the test length is doubled. (c) What percent improvement is there in the reliability when the test length is doubled? 53. Area If 100 feet of fence is to be used to enclose a rectangular yard, then the resulting area of the fenced yard is given by A ⫽ x(50 ⫺ x) where x is the width of the rectangle. (a) Is A a function of x? (b) If A ⫽ A(x), ﬁnd A(2) and A(30). (c) What restrictions must be placed on x (the domain) so that the problem makes physical sense? 54. Postal restrictions If a box with a square cross section is to be sent by a delivery service, there are restrictions on its size such that its volume is given by V ⫽ x2(108 ⫺ 4x), where x is the length of each side of the cross section (in inches). (a) Is V a function of x? (b) If V ⫽ V(x), ﬁnd V(10) and V(20). (c) What restrictions must be placed on x (the domain) so that the problem makes physical sense? 55. Proﬁt Suppose that the proﬁt from the production and sale of x units of a product is given by P(x) ⫽ 180x ⫺

x2 ⫺ 200 100

In addition, suppose that for a certain month the number of units produced on day t of the month is x ⫽ q(t) ⫽ 1000 ⫹ 10t (a) Find (P ⴰ q)(t) to express the proﬁt as a function of the day of the month. (b) Find the number of units produced, and the proﬁt, on the ﬁfteenth day of the month.

56. Fish species growth For many species of ﬁsh, the weight W is a function of the length L that can be expressed by W ⫽ W(L) ⫽ kL3

k ⫽ constant

Suppose that for a particular species k ⫽ 0.02, that for this species the length (in centimeters) is a function of the number of years t the ﬁsh has been alive, and that this function is given by L ⫽ L(t) ⫽ 50 ⫺

(t ⫺ 20)2 10

0 ⱕ t ⱕ 20

Find (W ⴰ L)(t) in order to express W as a function of the age t of the ﬁsh. 57. Revenue and advertising Suppose that a company’s revenue R ⫽ f(C ) is a function f of the number of customers C. Suppose also that the amount spent on advertising A affects the number of customers so that C ⫽ g(A) is a function g of A. (a) Is f ⴰ g deﬁned? Explain. (b) Is g ⴰ f deﬁned? Explain. (c) For the functions in (a) and (b) that are deﬁned, identify the input (independent variable) and the output (dependent variable) and explain what the function means. 58. Manufacturing Two of the processes (functions) used by a manufacturer of factory-built homes are sanding (denote this as function s) and painting (denote this as function p). Write a sentence of explanation for each of the following functional expressions involving s and p applied to a door. (a) s(door) (b) p(door) (c) ( p ⴰ s)(door) (d) (s ⴰ p)(door) (e) ( p ⴰ p)(door) 59. Fencing a lot A farmer wishes to fence the perimeter of a rectangular lot with an area of 1600 square feet. If the lot is x feet long, express the amount L of fence needed as a function of x. 60. Cost A shipping crate has a square base with sides of length x feet, and it is half as tall as it is wide. If the material for the bottom and sides of the box costs $2.00 per square foot and the material for the top costs $1.50 per square foot, express the total cost of material for the box as a function of x. 61. Revenue An agency charges $100 per person for a trip to a concert if 30 people travel in a group. But for each person above the 30, the amount charged each

1.3 Linear Functions ●

traveler will be reduced by $2.00. If x represents the number of people above the 30, write the agency’s revenue R as a function of x. 62. Revenue A company handles an apartment building with 50 units. Experience has shown that if the rent for

1.3

85

each of the units is $720 per month, all of the units will be ﬁlled, but one unit will become vacant for each $20 increase in the monthly rate. If x represents the number of $20 increases, write the revenue R from the building as a function of x.

Linear Functions

OBJECTIVES ● ● ●

●

●

●

To find the intercept of graphs To graph linear functions To find the slope of a line from its graph and from its equation To find the rate of change of a linear function To graph a line, given its slope and y-intercept or its slope and one point on the line To write the equation of a line, given information about its graph

Linear Function

] Application Preview The number of banks in the United States for selected years from 1980 to 2005 is given by y ⴝ ⴚ416.454x ⴙ 18,890.75 where x is the number of years after 1980. What does this function tell about how the number of banks has changed per year during this period? (See Example 6.) In this section, we will find the slopes and intercepts of graphs of linear functions and apply them.

The function in the Application Preview is an example of a special function, called the linear function. A linear function is deﬁned as follows. A linear function is a function of the form y ⫽ f (x) ⫽ ax ⫹ b where a and b are constants.

Intercepts

Intercepts

Because the graph of a linear function is a line, only two points are necessary to determine its graph. It is frequently possible to use intercepts to graph a linear function. The point(s) where a graph intersects the x-axis are called the x-intercept points, and the x-coordinates of these points are the x-intercepts. Similarly, the points where a graph intersects the y-axis are the y-intercept points, and the y-coordinates of these points are the y-intercepts. Because any point on the x-axis has y-coordinate 0 and any point on the y-axis has x-coordinate 0, we ﬁnd intercepts as follows. (a) To ﬁnd the y-intercept(s) of the graph of an equation, set x ⫽ 0 in the equation and solve for y. Note: A function of x has at most one y-intercept. (b) To ﬁnd the x-intercept(s), set y ⫽ 0 and solve for x.

● EXAMPLE 1 Intercepts Find the intercepts and graph the following. (a) 3x ⫹ y ⫽ 9 (b) x ⫽ 4y

86

● Chapter 1

Linear Equations and Functions

Solution (a) To ﬁnd the y-intercept, we set x ⫽ 0 and solve for y. 3(0) ⫹ y ⫽ 9 gives y ⫽ 9, so the y-intercept is 9. To ﬁnd the x-intercept, we set y ⫽ 0 and solve for x. 3x ⫹ 0 ⫽ 9 gives x ⫽ 3, so the x-intercept is 3. Using the intercepts gives the graph, shown in Figure 1.14. (b) Letting x ⫽ 0 gives y ⫽ 0, and letting y ⫽ 0 gives x ⫽ 0, so the only intercept of the graph of x ⫽ 4y is at the point (0, 0). A second point is needed to graph the line. Hence, if we let y ⫽ 1 in x ⫽ 4y, we get x ⫽ 4 and have a second point (4, 1) on the graph. It is wise to plot a third point as a check. The graph is shown in Figure 1.15. y

y 4

10

3x + y = 9

(0, 9)

3

x = 4y

2

5

(4,1)

1

(3, 0) x −10 −8 −6 −4 −2

2

4

6

8 10

x −4

−1 −1

−5

1

2

3

4

(0, 0)

−2 −3

−10

−4

Figure 1.14

Figure 1.15

Note that the equation graphed in Figure 1.14 can be rewritten as y ⫽ 9 ⫺ 3x or f(x) ⫽ 9 ⫺ 3x We see in Figure 1.14 that the x-intercept (3, 0) is the point where the function value is zero. The x-coordinate of such a point is called a zero of the function. Thus we see that the x-intercepts of a function are the same as its zeros.

● EXAMPLE 2 Depreciation A business property is purchased for $122,880 and depreciated over a period of 10 years. Its value y is related to the number of months of service x by the equation 4096x ⫹ 4y ⫽ 491,520 Find the x-intercept and the y-intercept and use them to sketch the graph of the equation. Solution x-intercept: y ⫽ 0

gives 4096x ⫽ 491,520 x ⫽ 120

Thus 120 is the x-intercept. y-intercept: x ⫽ 0

gives 4y ⫽ 491,520 y ⫽ 122,880

Thus 122,880 is the y-intercept. The graph is shown in Figure 1.16. Note that the units on the x- and y-axes are different and that the y-intercept corresponds to the value of the property 0 months after purchase. That is, the y-intercept gives the purchase price. The x-intercept corresponds to the number of months that have passed before the value is 0; that

1.3 Linear Functions ●

87

is, the property is fully depreciated after 120 months, or 10 years. Note that only positive values for x and y make sense in this application, so only the Quadrant I portion of the graph is shown. y 140,000 120,000

4096x + 4y = 491,520

Dollars

100,000 80,000 60,000 40,000 20,000 x 20

40

Figure 1.16

60

80

100 120 140

Months

Despite the ease of using intercepts to graph linear equations, this method is not always the best. For example, vertical lines, horizontal lines, or lines that pass through the origin may have a single intercept, and if a line has both intercepts very close to the origin, using the intercepts may lead to an inaccurate graph.

Rate of Change; Slope of a Line

Note that in Figure 1.16, as the graph moves from the y-intercept point (0, 122,880) to the x-intercept point (120, 0), the y-value on the line changes ⫺122,880 units (from 122,880 to 0), whereas the x-value changes 120 units (from 0 to 120). Thus the rate of change of the value of the business property is ⫺122,880 ⫽ ⫺1024 dollars per month 120 This means that each month the value of the property changes by ⫺1024 dollars, or the value decreases by $1024/month. This rate of change of a linear function is called the slope of the line that is its graph (see Figure 1.16). For the graph of a linear function, the ratio of the change in y to the corresponding change in x measures the slope of the line. For any nonvertical line, the slope can be found by using any two points on the line, as follows.

Slope of a Line

If a nonvertical line passes through the points P1(x1, y1) and P2(x2, y2), its slope, denoted by m, is found by using either m⫽

y2 ⫺ y1 x2 ⫺ x1

or, equivalently,

y

P2(x2, y2)

y2

y2 − y1

P1(x1, y1)

m⫽

y1 ⫺ y2

y1

x2 − x1

x1 ⫺ x2

The slope of a vertical line is undeﬁned.

x x1

x2

Note that for a given line, the slope is the same regardless of which two points are used in the calculation; this is because corresponding sides of similar triangles are in proportion.

88

● Chapter 1

Linear Equations and Functions

We may also write the slope by using the notation m⫽

¢y ¢x

(¢y ⫽ y2 ⫺ y1

and

¢x ⫽ x2 ⫺ x1)

where ¢y is read “delta y” and means “change in y,” and ¢x means “change in x.”

● EXAMPLE 3 Slopes Find the slope of (a) line /1, passing through (⫺2, 1) and (4, 3) (b) line /2, passing through (3, 0) and (4, ⫺3) Solution 1⫺3 3⫺1 2 1 ⫺2 1 ⫽ ⫽ or, equivalently, m ⫽ ⫽ ⫽ 4 ⫺ (⫺2) 6 3 ⫺2 ⫺ 4 ⫺6 3 This means that a point 3 units to the right and y 1 unit up from any point on the line is also on the 4 line. Line /1 is shown in Figure 1.17.

(a) m ⫽

0 ⫺ (⫺3) 3 ⫽ ⫽ ⫺3 (b) m ⫽ 3⫺4 ⫺1 This means that a point 1 unit to the right and 3 units down from any point on the line is also on the line. Line /2 is also shown in Figure 1.17.

(4, 3) ℓ1

2

(−2, 1)

(3, 0) x −3 −2 −1

1

2

3

4

5

6

−2

(4, −3)

Figure 1.17

−4

ℓ2

From the previous discussion, we see that the slope describes the direction of a line as follows.

Orientation of a Line and Its Slope 1. The slope is positive if the line slopes upward toward the right. The function is increasing.

3. The slope of a horizontal line is 0, because ¢y ⫽ 0. The function is constant. y

y

m⫽

¢y ⬎0 ¢x

m>0 x

2. The slope is negative if the line slopes downward toward the right. The function is decreasing.

m⫽

¢y ⫽0 ¢x

¢y 6 0 ¢x

x

4. The slope of a vertical line is undeﬁned, because ¢x ⫽ 0. y

y

m⫽

m=0

x

mg)(x) (c) f(g(x)) (d) ( f ⴰ f )(x) In Problems 24–26, ﬁnd the intercepts and graph. 24. 5x ⫹ 2y ⫽ 10 25. 6x ⫹ 5y ⫽ 9 26. x ⫽ ⫺2 In Problems 27 and 28, ﬁnd the slope of the line that passes through each pair of points. 27. (2, ⫺1) and (⫺1, ⫺4) 28. (⫺3.8, ⫺7.16) and (⫺3.8, 1.16) In Problems 29 and 30, ﬁnd the slope and y-intercept of each line. 29. 2x ⫹ 5y ⫽ 10 3 3 30. x ⫽ ⫺ y ⫹ 4 2

4

Figure 1.51

20. For the function f graphed in Figure 1.51, what is f(2)? 21. For the function f graphed in Figure 1.51, for what values of x does f(x) ⫽ 0? 22. The following table deﬁnes y as a function of x, denoted y ⫽ f(x).

4

In Problems 31–37, write the equation of each line described. 31. Slope 4 and y-intercept 2 32. Slope ⫺12 and y-intercept 3 33. Through (⫺2, 1) with slope 25 34. Through (⫺2, 7) and (6, ⫺4) 35. Through (⫺1, 8) and (⫺1, ⫺1) 36. Through (1, 6) and parallel to y ⫽ 4x ⫺ 6 37. Through (⫺1, 2) and perpendicular to 3x ⫹ 4y ⫽ 12 In Problems 38 and 39, graph each equation with a graphing utility and the standard viewing window. 38. x2 ⫹ y ⫺ 2x ⫺ 3 ⫽ 0 x3 ⫺ 27x ⫹ 54 39. y ⫽ 15

Review Exercises ●

cost him $0.30 per mile to operate. Find the number of miles he must drive before the costs are equal. If he normally keeps a truck for 5 years, which is the better buy? 54. Heart disease risk The Multiple Risk Factor Intervention Trial (MRFIT) used data from 356,222 men aged 35 to 57 to investigate the relationship between serum cholesterol and coronary heart disease (CHD) risk. Figure 1.53 shows the graph of the relationship of CHD risk and cholesterol, where a risk of 1 is assigned to 200 mg/dl of serum cholesterol and where the CHD risk is 4 times as high when serum cholesterol is 300 mg/dl.

In Problems 44–50, solve each system of equations. 4x ⫺ 2y ⫽ 6 2x ⫹ y ⫽ 19 44. b 45. b 3x ⫹ 3y ⫽ 9 x ⫺ 2y ⫽ 12 3x ⫹ 2y ⫽ 5 6x ⫹ 3y ⫽ 1 46. b 47. b 2x ⫺ 3y ⫽ 12 y ⫽ ⫺2x ⫹ 1 4x ⫺ 3y ⫽ 253 13x ⫹ 2y ⫽ ⫺12 x ⫹ 2y ⫹ 3z ⫽ 5 49. c y ⫹ 11z ⫽ 21 5y ⫹ 9z ⫽ 13

x ⫹ y ⫺ z ⫽ 12 50. c 2y ⫺ 3z ⫽ ⫺7 3x ⫹ 3y ⫺ 7z ⫽ 0

A P P L I C AT I O N S 51. Endangered animals The number of species of endangered animals, A, can be described by A ⫽ 9.78x ⫹ 167.90 where x is the number of years past 1980. (Source: U.S. Fish and Wildlife Service) (a) To what year does x ⫽ 17 correspond? (b) What x-value corresponds to the year 2007? (c) If this equation remains valid, in what year will the number of species of endangered animals reach 461? 52. Course grades In a certain course, grades are based on three tests worth 100 points each, three quizzes worth 50 points each, and a ﬁnal exam worth 200 points. A student has test grades of 91, 82, and 88, and quiz grades of 50, 42, and 42. What is the lowest percent the student can get on the ﬁnal and still earn an A (90% or more of the total points) in the course? 53. Cost analysis The owner of a small construction business needs a new truck. He can buy a diesel truck for $38,000 and it will cost him $0.24 per mile to operate. He can buy a gas engine truck for $35,600 and it will

3.0

2.0

2.0 1.0

0.7 1.0

0 100 150 200 250 300

Figure 1.53

48. b

4.0

4.0

CHD risk ratio

In Problems 40 and 41, use a graphing utility and (a) graph each equation in the viewing window given. (b) graph each equation in the standard viewing window. (c) explain how the two views differ and why. 40. y ⫽ (x ⫹ 6)(x ⫺ 3)(x ⫺ 15) with x-min ⫽ ⫺15, x-max ⫽ 25; y-min ⫽ ⫺700, y-max ⫽ 500 41. y ⫽ x2 ⫺ x ⫺ 42 with x-min ⫽ ⫺15, x-max ⫽ 15; y-min ⫽ ⫺50, y-max ⫽ 50 1x ⫹ 3 42. What is the domain of y ⫽ ? Check with a x graphing utility. 43. Use a graphing utility and the x-intercept method to approximate the solution of 7x ⫺ 2 ⫽ 0.

133

Serum cholesterol (mg/dl)

(a) Does this graph indicate that the CHD risk is a function of the serum cholesterol? (b) Is the relationship a linear function? (c) If CHD risk is a function f of serum cholesterol, what is f(300)? 55. Mortgage loans When a couple purchases a home, one of the ﬁrst questions they face deals with the relationship between the amount borrowed and the monthly payment. In particular, if a bank offers 25-year loans at 7% interest, then the data in the following table would apply. Amount Borrowed

Monthly Payment

$40,000 50,000 60,000 70,000 80,000 90,000 100,000

$282.72 353.39 424.07 494.75 565.44 636.11 706.78

Source: Comprehensive Mortgage Payment Tables, Publication No. 492, Financial Publishing Co., Boston

Assume that the monthly payment P is a function of the amount borrowed A (in thousands) and is denoted by P ⫽ f(A), and answer the following. (a) Find f(80). (b) Write a sentence that explains the meaning of f(70) ⫽ 494.75.

134

● Chapter 1

Linear Equations and Functions

56. Proﬁt Suppose that the proﬁt from the production and sale of x units of a product is given by P(x) ⫽ 330x ⫺ 0.05x2 ⫺ 5000 In addition, suppose that for a certain month, the number of units produced on day t of the month is x ⫽ q(t) ⫽ 100 ⫹ 10t

57. Fish species growth For many species of ﬁsh, the weight W is a function of the length L that can be expressed by k ⫽ constant

Suppose that for a particular species k ⫽ 0.03 and that for this species, the length (in centimeters) is a function of the number of years t the ﬁsh has been alive and that this function is given by L ⫽ L(t) ⫽ 65 ⫺ 0.1(t ⫺ 25)2

0 ⱕ t ⱕ 25

Find (W ⴰ L)(t) in order to express W as a function of the age t of the ﬁsh. 58. Distance to a thunderstorm The distance d (in miles) to a thunderstorm is given by d⫽

61. Health care costs The average annual cost per consumer for health care can be modeled by A ⫽ 71.72x ⫹ 1401.36

(a) Find (P ⴰ q)(t) to express the proﬁt as a function of the day of the month. (b) Find the number of units produced, and the proﬁt, on the ﬁfteenth day of the month.

W ⫽ W(L) ⫽ kL3

(a) If 200 units sold results in $3100 proﬁt and 250 units sold results in $6000 proﬁt, write the proﬁt function for this company. (b) Interpret the slope from part (a) as a rate of change.

t 4.8

where t is the number of seconds that elapse between seeing the lightning and hearing the thunder. (a) Graph this function for 0 ⱕ t ⱕ 20. (b) The point (9.6, 2) satisﬁes the equation. Explain its meaning. 59. Body-heat loss Body-heat loss due to convection depends on a number of factors. If Hc is body-heat loss due to convection, Ac is the exposed surface area of the body, Ts ⫺ Ta is skin temperature minus air temperature, and Kc is the convection coefﬁcient (determined by air velocity and so on), then we have Hc ⫽ KcAc(Ts ⫺ Ta) When Kc ⫽ 1, Ac ⫽ 1, and Ts ⫽ 90, the equation is Hc ⫽ 90 ⫺ Ta Sketch the graph. 60. Proﬁt A company charting its proﬁts notices that the relationship between the number of units sold, x, and the proﬁt, P, is linear.

where x is the number of years from 1990. (Source: www.infoplease.com) (a) Is A a linear function of x? (b) Find the slope and A-intercept of this function. (c) Write a sentence that interprets the A-intercept. (d) Write a sentence that interprets the slope of this function as a rate of change. 62. Temperature Write the equation of the linear relationship between temperature in Celsius (C) and Fahrenheit (F) if water freezes at 0°C and 32°F and boils at 100°C and 212°F. 63. Photosynthesis The amount y of photosynthesis that takes place in a certain plant depends on the intensity x of the light present, according to y ⫽ 120x2 ⫺ 20x3

for x ⱖ 0

(a) Graph this function with a graphing utility. (Use y-min ⫽ ⫺100 and y-max ⫽ 700.) (b) The model is valid only when f(x) ⱖ 0 (that is, on or above the x-axis). For what x-values is this true? 64. Flow rates of water The speed at which water travels in a pipe can be measured by directing the ﬂow through an elbow and measuring the height to which it spurts out the top. If the elbow height is 10 cm, the equation relating the height h (in centimeters) of the water above the elbow and its velocity v (in centimeters per second) is given by v2 ⫽ 1960(h ⫹ 10) (a) Solve this equation for h and graph the result, using the velocity as the independent variable. (b) If the velocity is 210 cm/sec, what is the height of the water above the elbow? 65. Investment mix A retired couple have $150,000 to invest and want to earn $15,000 per year in interest. The safer investment yields 9.5%, but they can supplement their earnings by investing some of their money at 11%. How much should they invest at each rate to earn $15,000 per year?

Chapter Test ●

66. Botany A botanist has a 20% solution and a 70% solution of an insecticide. How much of each must be used to make 4.0 liters of a 35% solution? 67. Supply and demand A certain product has supply and demand functions p ⫽ 4q ⫹ 5 and p ⫽ ⫺2q ⫹ 81, respectively. (a) If the price is $53, how many units are supplied and how many are demanded? (b) Does this give a shortfall or a surplus? (c) Is the price likely to increase from $53 or decrease from it? 68. Market analysis Of the equations p ⫹ 6q ⫽ 420 and p ⫽ 6q ⫹ 60, one is the supply function for a product and one is the demand function for that product. (a) Graph these equations on the same set of axes. (b) Label the supply function and the demand function. (c) Find the market equilibrium point. 69. Cost, revenue, and proﬁt The total cost and total revenue for a certain product are given by the following: C(x) ⫽ 38.80x ⫹ 4500 R(x) ⫽ 61.30x (a) (b) (c) (d)

Find Find Find Find

the the the the

marginal cost. marginal revenue. marginal proﬁt. number of units required to break even.

70. Cost, revenue, and proﬁt A certain commodity has the following costs for a period. Fixed costs: Variable costs:

1

The commodity (a) What is the (b) What is the (c) What is the (d) What is the (e) What is the (f) What is the (g) What is the

135

is sold for $52 per unit. total cost function? total revenue function? proﬁt function? marginal cost? marginal revenue? marginal proﬁt? break-even point?

71. Market analysis The supply function and the demand function for a product are linear and are determined by the tables that follow. Find the quantity and price that will give market equilibrium. Supply Function

Demand Function

Price

Quantity

Price

Quantity

100 200 300

200 400 600

200 100 0

200 400 600

72. Market analysis Suppose that for a certain product the supply and demand functions prior to any taxation are Supply: Demand:

q ⫹8 10 10p ⫹ q ⫽ 1500 p⫽

If a tax of $2 per item is levied on the supplier and is passed on to the consumer as a price increase, ﬁnd the market equilibrium after the tax is levied.

$1500 $22 per unit

Chapter Test

In Problems 1–3, solve the equations. x 1. 4x ⫺ 3 ⫽ ⫹ 6 2 4x 3 3x ⫺ 1 5 2. ⫹ 4 ⫽ 3. ⫽ x x⫹1 4x ⫺ 9 7 4. For f(x) ⫽ 7 ⫹ 5x ⫺ 2x2, ﬁnd and simplify f(x ⫹ h) ⫺ f(x) . h 2 5. Solve 1 ⫹ t ⱕ 3t ⫹ 22 and graph the solution. 3 In Problems 6 and 7, ﬁnd the intercepts and graph the functions. 6. 5x ⫺ 6y ⫽ 30 7. 7x ⫹ 5y ⫽ 21

8. Consider the function f(x) ⫽ 14x ⫹ 16. (a) Find the domain and range. (b) Find f(3). (c) Find the y-coordinate on the graph of y ⫽ f (x) when x ⫽ 5. 9. Write the equation of the line passing through (⫺1, 2) and (3, ⫺4). Write your answer in slope-intercept form. 10. Find the slope and the y-intercept of the graph of 5x ⫹ 4y ⫽ 15. 11. Write the equation of the line through (⫺3, ⫺1) that (a) has undeﬁned slope. (b) is perpendicular to x ⫽ 4y ⫺ 8.

136

● Chapter 1

Linear Equations and Functions

12. Which of the following relations [(a), (b), (c)] are functions? Explain. y (a)

x

(c) y ⫽ ⫾ 1x2 ⫺ 1 x y 1 3 ⫺1 3 4 2 13. Solve the system (b)

b

3x ⫹ 2y ⫽ ⫺2 4x ⫹ 5y ⫽ 2

14. Given f(x) ⫽ 5x2 ⫺ 3x and g(x) ⫽ x ⫹ 1, ﬁnd (a) ( fg)(x) (b) g(g(x)) (c) ( f ⴰ g)(x) 15. The total cost function for a product is C(x) ⫽ 30x ⫹ 1200, and the total revenue is R(x) ⫽ 38x, where x is the number of units produced and sold. (a) Find the marginal cost. (b) Find the proﬁt function. (c) Find the number of units that gives break-even. (d) Find the marginal proﬁt and explain what it means. ˛

16. The selling price for each item of a product is $50, and the total cost is given by C(x) ⫽ 10x ⫹ 18,000, where x is the number of items. (a) Write the revenue function. (b) Find C(100) and write a sentence that explains its meaning. (c) Find the number of units that gives the break-even point. 17. The supply function for a product is p ⫽ 5q ⫹ 1500 and the demand function is p ⫽ ⫺3q ⫹ 3100. Find the quantity and price that give market equilibrium. 18. A building is depreciated by its owner, with the value y of the building after x months given by y ⫽ 720,000 ⫺ 2000x. (a) Find the y-intercept of the graph of this function, and explain what it means. (b) Find the slope of the graph and tell what it means. 19. An airline has 360 seats on a plane for one of its ﬂights. If 90% of the people making reservations actually buy a ticket, how many reservations should the airline accept to be conﬁdent that it will sell 360 tickets? 20. Amanda plans to invest $20,000, part of it at a 9% interest rate and part of it in a safer fund that pays 6%. How much should be invested in each fund to yield an annual return of $1560?

Extended Applications & Group Projects I. Hospital Administration Southwest Hospital has an operating room used only for eye surgery. The annual cost of rent, heat, and electricity for the operating room and its equipment is $360,000, and the annual salaries of the people who staff this room total $540,000. Each surgery performed requires the use of $760 worth of medical supplies and drugs. To promote goodwill, every patient receives a bouquet of ﬂowers the day after surgery. In addition, one-quarter of the patients require dark glasses, which the hospital provides free of charge. It costs the hospital $30 for each bouquet of ﬂowers and $40 for each pair of glasses. The hospital receives a payment of $2000 for each eye operation performed. 1. Identify the revenue per case and the annual ﬁxed and variable costs for running the operating room. 2. How many eye operations must the hospital perform each year in order to break even? 3. Southwest Hospital currently averages 70 eye operations per month. One of the nurses has just learned about a machine that would reduce by $100 per patient the amount of medical supplies needed. It can be leased for $100,000 annually. Keeping in mind the ﬁnancial cost and beneﬁts, advise the hospital on whether it should lease this machine. 4. An advertising agency has proposed to the hospital’s president that she spend $20,000 per month on television and radio advertising to persuade people that Southwest Hospital is the best place to have any eye surgery performed. Advertising account executives estimate that such publicity would increase business by 40 operations per month. If they are correct and if this increase is not big enough to affect ﬁxed costs, what impact would this advertising have on the hospital’s proﬁts? 5. In case the advertising agency is being overly optimistic, how many extra operations per month are needed to cover the cost of the proposed ads? 6. If the ad campaign is approved and subsequently meets its projections, should the hospital review its decision about the machine discussed in Question 3?

137

138

● Chapter 1

Linear Equations and Functions

II. Fundraising At most colleges and universities, student organizations conduct fundraising activities, such as selling T-shirts or candy. If a club or organization decided to sell coupons for submarine sandwiches, it would want to ﬁnd the best deal, based on the amount the club thought it could sell and how much proﬁt it would make. In this project, you’ll try to discover the best deal for this type of fundraiser conducted in your area. 1. Contact at least two different sub shops in your area from among local sub shops, a national chain, or a regional or national convenience store chain. From each contact, ﬁnd the details of selling sub sandwich coupons as a fundraiser. In particular, determine the following for each sub shop, and present your ﬁndings in a chart. (a) The selling price for each coupon and its value to the coupon holder, including any expiration date (b) Your cost for each coupon sold and for each one returned (c) The total number of coupons provided (d) The duration of the sale 2. For each sub shop, determine a club’s total revenue and total costs as linear functions of the number of coupons sold. 3. Form the proﬁt function for each sub shop, and graph the proﬁt functions together. For each function, determine the break-even point. Find the marginal proﬁt for each function, and interpret its meaning. 4. The shop with the best deal will be the one whose coupons will provide the maximum proﬁt for a club. (a) For each shop, determine a sales estimate that is based on location, local popularity, and customer value. (b) Use each estimate from (a) with that shop’s proﬁt function to determine which shop gives the maximum proﬁt. Fully explain and support your claims; make and justify a recommendation.

2 Quadratic and Other Special Functions In Chapter 1 we discussed functions in general and linear functions in particular. In this chapter we will discuss quadratic functions and their applications, and we will also discuss other types of functions, including identity, constant, power, absolute value, piecewise defined, and reciprocal functions. Graphs of polynomial and rational functions will also be introduced; they will be studied in detail in Chapter 10. We will pay particular attention to quadratic equations and to quadratic functions, and we

Sections

Applications

2.1 Quadratic Equations Factoring methods The quadratic formula

Social Security benefits, falling objects

2.2 Quadratic Functions: Parabolas Vertices of parabolas Zeros

Profit maximization, maximizing revenue

2.3 Business Applications of Quadratic Functions

Supply, demand, market equilibrium, break-even points, profit maximization

2.4 Special Functions and Graphs Basic functions Polynomial functions Rational functions Piecewise defined functions

Selling price, residential electric costs, average cost

2.5 Modeling; Fitting Curves to Data Linear regression

Internet access, federal tax per capita, expected life span, consumer price index

will see that cost, revenue, profit, supply, and demand are sometimes modeled by quadratic functions. We also include, in an optional section on modeling, the creation of functions that approximately fit real data points. The topics and some of the applications discussed in this chapter follow.

139

Chapter Warm-up Prerequisite Problem Type

Answer

Section for Review

(a) 12 (b) 1 (c) ⫺15

0.2 Signed numbers

2.1 2.2 2.3

(a) (3x ⫺ 2)(2x ⫹ 1) (b) 3x(2x ⫺ 3)

0.6 Factoring

2.1

No

0.4 Radicals

(a) Find the y-intercept of y ⫽ x2 ⫺ 6x ⫹ 8. (b) Find the x-intercept of y ⫽ 2x ⫺ 3.

2.2 2.3

(a) 8

1.3 x- and y-intercepts

If f(x) ⫽ ⫺x2 ⫹ 4x, ﬁnd f(2). Find the domain of

2.2 2.3 2.4

Find b2 ⫺ 4ac if (a) a ⫽ 1, b ⫽ 2, c ⫽ ⫺2 (b) a ⫽ ⫺2, b ⫽ 3, c ⫽ ⫺1 (c) a ⫽ ⫺3, b ⫽ ⫺3, c ⫽ ⫺2 (a) Factor 6x2 ⫺ x ⫺ 2. (b) Factor 6x2 ⫺ 9x.

Is

1 ⫹ 1⫺3 a real number? 2

f(x) ⫽

12x ⫹ 8 . 3x ⫺ 9

Assume revenue is R(x) ⫽ 500x ⫺ 2x2 and cost is C(x) ⫽ 3600 ⫹ 100x ⫹ 2x2. (a) Find the proﬁt function P(x). (b) Find P(50).

140

For Section 2.1 2.2 2.3

(b)

3 2

4

1.2 Functions

All x ⫽ 3 1.6 Cost, revenue, and proﬁt

2.3

(a) P(x) ⫽ ⫺3600 ⫹ 400x ⫺ 4x2 (b) P(50) ⫽ 6400

2.1 Quadratic Equations ●

2.1

141

Quadratic Equations

OBJECTIVES ●

●

To solve quadratic equations with factoring methods To solve quadratic equations with the quadratic formula

] Application Preview With fewer workers, future income from payroll taxes used to fund Social Security benefits won’t keep pace with scheduled benefits. The function B ⴝ ⴚ1.056785714t2 ⴙ 8.259285714t ⴙ 74.07142857 describes the Social Security trust fund balance B, in billions of dollars, where t is the number of years past the year 2000 (Source: Social Security Administration). For planning purposes, it is important to know when the trust fund balance will be 0. That is, for what t-value does 0 ⴝ ⴚ1.056785714t2 ⴙ 8.259285714t ⴙ 74.07142857? (See Example 7 for the solution.) In this section, we learn how to solve equations of this type, using factoring methods and using the quadratic formula.

Factoring Methods It is important for us to know how to solve equations, such as the equation in the Application Preview. This was true for linear equations and is again true for quadratic equations. A quadratic equation in one variable is an equation that can be written in the general form ax2 ⫹ bx ⫹ c ⫽ 0

(a ⫽ 0)

where a, b, and c represent constants. For example, the equations 3x2 ⫹ 4x ⫹ 1 ⫽ 0

and

2x2 ⫹ 1 ⫽ x2 ⫺ x

are quadratic equations; the ﬁrst of these is in general form, and the second can easily be rewritten in general form. When we solve quadratic equations, we will be interested only in real number solutions and will consider two methods of solution: factoring and the quadratic formula. We will discuss solving quadratic equations by factoring ﬁrst. (For a review of factoring, see Section 0.6, “Factoring.”) Solution by factoring is based on the following property of the real numbers. Zero Product Property

For real numbers a and b, ab ⫽ 0 if and only if a ⫽ 0 or b ⫽ 0 or both. Hence, to solve by factoring, we must ﬁrst write the equation with zero on one side.

● EXAMPLE 1 Solving Quadratic Equations Solve: (a) 6x2 ⫹ 3x ⫽ 4x ⫹ 2

(b) 6x2 ⫽ 9x

142

● Chapter 2

Quadratic and Other Special Functions

Solution (a)

6x2 ⫹ 3x ⫽ 4x ⫹ 2 ⫺x⫺2⫽0 (3x ⫺ 2)(2x ⫹ 1) ⫽ 0 3x ⫺ 2 ⫽ 0 or 2x ⫹ 1 ⫽ 0 3x ⫽ 2 2x ⫽ ⫺1 2 1 x⫽ x⫽⫺ 3 2 6x2

Proper form for factoring Factored Factors equal to zero

Solutions

We now check that these values are, in fact, solutions to our original equation. 2 2 2 2 6a b ⫹ 3a b ⱨ 4a b ⫹ 2 3 3 3

1 2 1 1 6a⫺ b ⫹ 3a⫺ b ⱨ 4a⫺ b ⫹ 2 2 2 2

14 14 ⫽ ✔ 3 3

0⫽0✔

6x2 ⫽ 9x ⫺ 9x ⫽ 0 3x(2x ⫺ 3) ⫽ 0 3x ⫽ 0 or 2x ⫺ 3 ⫽ 0 x⫽0 2x ⫽ 3 3 x⫽ 2

(b)

6x2

Check: 6(0)2 ⫽ 9(0) ✔

3 2 3 6a b ⫽ 9a b ✔ 2 2

Thus the solutions are x ⫽ 0 and x ⫽ 23.

Note that in Example 1(b) it is tempting to divide both sides of the equation by x, but this is incorrect because it results in the loss of the solution x ⫽ 0. Never divide both sides of an equation by an expression containing the variable.

● EXAMPLE 2 Solving by Factoring Solve: (a) (y ⫺ 3)(y ⫹ 2) ⫽ ⫺4 for y

(b)

x⫹1 3 2x ⫹ 6 ⫽ ⫹ x 3x ⫹ 6 x(3x ⫹ 6)

Solution (a) Note that the left side of the equation is factored, but the right member is not 0. Therefore, we must multiply the factors before we can rewrite the equation in general form. (y ⫺ 3)(y ⫹ 2) ⫽ ⫺4 y2 ⫺ y ⫺ 6 ⫽ ⫺4 y2 ⫺ y ⫺ 2 ⫽ 0 (y ⫺ 2)(y ⫹ 1) ⫽ 0 y ⫺ 2 ⫽ 0 or y ⫹ 1 ⫽ 0 y⫽2 y ⫽ ⫺1 Check: (2 ⫺ 3)(2 ⫹ 2) ⫽ ⫺4 ✔

(⫺1 ⫺ 3)(⫺1 ⫹ 2) ⫽ ⫺4 ✔

2.1 Quadratic Equations ●

143

(b) The LCD of all fractions is x(3x ⫹ 6). Multiplying both sides of the equation by this LCD gives a quadratic equation that is equivalent to the original equation for x ⫽ 0 and x ⫽ ⫺2. (The original equation is undeﬁned for these values.) 2x ⫹ 6 x⫹1 3 ⫽ ⫹ x 3x ⫹ 6 x(3x ⫹ 6) 2x ⫹ 6 x⫹1 3 x(3x ⫹ 6) ⫽ x(3x ⫹ 6)¢ ⫹ ≤ x 3x ⫹ 6 x(3x ⫹ 6) x(x ⫹ 1) ⫽ 3(3x ⫹ 6) ⫹ (2x ⫹ 6) x2 ⫹ x ⫽ 9x ⫹ 18 ⫹ 2x ⫹ 6 x2 ⫺ 10x ⫺ 24 ⫽ 0 (x ⫺ 12)(x ⫹ 2) ⫽ 0 x ⫺ 12 ⫽ 0 or x ⫹ 2 ⫽ 0 x ⫽ 12 or x ⫽ ⫺2

Multiply both sides by x(3x ⫹ 6).

Checking x ⫽ 12 and x ⫽ ⫺2 in the original equation, we see that x ⫽ ⫺2 does not check because it makes the denominator equal to zero. Hence the only solution is x ⫽ 12. ✔ ●

Checkpoint

1. The factoring method for solving a quadratic equation is based on the __________ product property. Hence, in order for us to solve a quadratic equation by factoring, one side of the equation must equal _________. 2. Solve the following equations by factoring. (a) x2 ⫺ 19x ⫽ 20 (b) 2x2 ⫽ 6x

● EXAMPLE 3 Falling Object A tennis ball is thrown into a swimming pool from the top of a tall hotel. The height of the ball from the pool is modeled by D(t) ⫽ ⫺16t2 ⫺ 4t ⫹ 300 feet, where t is the time, in seconds, after the ball was thrown. How long after the ball was thrown was it 144 feet above the pool? Solution To ﬁnd the number of seconds until the ball is 144 feet above the pool, we solve the equation 144 ⫽ ⫺16t2 ⫺ 4t ⫹ 300 for t. The solution follows. 144 ⫽ ⫺16t2 ⫺ 4t ⫹ 300 0 ⫽ ⫺16t2 ⫺ 4t ⫹ 156 0 ⫽ 4t2 ⫹ t ⫺ 39 0 ⫽ (t ⫺ 3)(4t ⫹ 13) t ⫺ 3 ⫽ 0 or 4t ⫹ 13 ⫽ 0 t ⫽ 3 or t ⫽ ⫺13兾4 This indicates that the ball will be 144 feet above the pool 3 seconds after it is thrown. The negative value for t has no meaning in this application.

The Quadratic Formula Factoring does not lend itself easily to solving quadratic equations such as x2 ⫺ 5 ⫽ 0

144

● Chapter 2

Quadratic and Other Special Functions

However, we can solve this equation by writing x2 ⫽ 5 x ⫽ ⫾ 15 In general, we can solve quadratic equations of the form x2 ⫽ C (no x-term) by taking the square root of both sides. Square Root Property

x ⫽ ⫾ 1C

The solution of x2 ⫽ C is

This property also can be used to solve equations such as those in the following example.

● EXAMPLE 4 Square Root Method Solve the following equations. (a) 4x2 ⫽ 5

(b) (3x ⫺ 4)2 ⫽ 9

Solution We can use the square root property for both parts. (a) 4x2 ⫽ 5 is equivalent to x2 ⫽ 54. Thus x⫽⫾

5 15 15 ⫽⫾ ⫽⫾ B4 2 14

(b) (3x ⫺ 4)2 ⫽ 9 is equivalent to 3x ⫺ 4 ⫽ ⫾ 19. Thus 3x ⫺ 4 ⫽ 3 3x ⫽ 7 7 x⫽ 3

3x ⫺ 4 ⫽ ⫺3 3x ⫽ 1 1 x⫽ 3

The solution of the general quadratic equation ax2 ⫹ bx ⫹ c ⫽ 0, where a ⫽ 0, is called the quadratic formula. It can be derived by using the square root property, as follows: ax2 ⫹ bx ⫹ c ⫽ 0 ax2 ⫹ bx ⫽ ⫺c c b x2 ⫹ x ⫽ ⫺ a a

Standard form Subtract c from both sides. Divide both sides by a.

We would like to make the left side of the last equation a perfect square of the form (x ⫹ k)2 ⫽ x2 ⫹ 2kx ⫹ k2. If we let 2k ⫽ b兾a, then k ⫽ b兾(2a) and k2 ⫽ b2兾(4a2). Hence we continue as follows: b2 b b2 c x2 ⫹ x ⫹ 2 ⫽ 2 ⫺ a a 4a 4a 2 2 b b ⫺ 4ac ¢x ⫹ ≤ ⫽ 2a 4a2 x⫹

b b2 ⫺ 4ac ⫽⫾ 2a A 4a2 ⫺b 1b2 ⫺ 4ac x⫽ ⫾ 2a 2兩a兩

Add

b2 to both sides. 4a2

Simplify.

Square root property

Solve for x.

2.1 Quadratic Equations ●

Quadratic Formula

Because ⫾2 0a 0 represents the same numbers as ⫾2a, we obtain the following.

145

⫺b ⫾ 1b2 ⫺ 4ac 2a

If ax2 ⫹ bx ⫹ c ⫽ 0, where a ⫽ 0, then x⫽

We may use the quadratic formula to solve all quadratic equations, but especially those in which factorization is difﬁcult or impossible to see. The proper identiﬁcation of values for a, b, and c to be substituted into the formula requires that the equation be in general form.

● EXAMPLE 5 Quadratic Formula Use the quadratic formula to solve 2x2 ⫺ 3x ⫺ 6 ⫽ 0 for x. Solution The equation is already in general form, with a ⫽ 2, b ⫽ ⫺3, and c ⫽ ⫺6. Hence ⫺b ⫾ 1b2 ⫺ 4ac 2a ⫺(⫺3) ⫾ 1(⫺3)2 ⫺ 4(2)(⫺6) ⫽ 2(2) 3 ⫾ 19 ⫹ 48 ⫽ 4 3 ⫾ 157 ⫽ 4

x⫽

Thus the solutions are x⫽

3 ⫹ 157 4

and x ⫽

3 ⫺ 157 4

● EXAMPLE 6 Nonreal Solution Using the quadratic formula, ﬁnd the (real) solutions to x2 ⫽ x ⫺ 1. Solution We must rewrite the equation in general form before we can determine the values of a, b, and c. x2 ⫽ x ⫺ 1 x2 ⫺ x ⫹ 1 ⫽ 0 Note: a ⫽ 1, b ⫽ ⫺1, and c ⫽ 1. ⫺(⫺1) ⫾ 1(⫺1)2 ⫺ 4(1)(1) x⫽ 2(1) 1 ⫾ 11 ⫺ 4 x⫽ 2 1 ⫾ 1⫺3 x⫽ 2

Because 1⫺3 is not a real number, the values of x are not real. Hence there are no real solutions to the given equation.

146

● Chapter 2

Quadratic and Other Special Functions

In Example 6 there were no real solutions to the quadratic equation because the radicand of the quadratic formula was negative. In general, when solving a quadratic equation, we can use the sign of the radicand in the quadratic formula—that is, the sign of b2 ⫺ 4ac —to determine how many real solutions there are. Thus we refer to b2 ⫺ 4ac as the quadratic discriminant. Quadratic Discriminant

Given ax2 ⫹ bx ⫹ c ⫽ 0 and a ⫽ 0, If b2 ⫺ 4ac ⬎ 0, the equation has two distinct real solutions. If b2 ⫺ 4ac ⫽ 0, the equation has exactly one real solution. If b2 ⫺ 4ac ⬍ 0, the equation has no real solutions. The quadratic formula is especially useful when the coefﬁcients of a quadratic equation are decimal values that make factorization impractical. This occurs in many applied problems. ] EXAMPLE 7 Social Security Trust Fund (Application Preview) The function B ⫽ ⫺1.056785714t2 ⫹ 8.259285714t ⫹ 74.07142857 describes the Social Security trust fund balance B, in billions of dollars, where t is the number of years past the year 2000 (Source: Social Security Administration). For planning purposes, it is important to know when the trust fund balance will be 0. To ﬁnd when this occurs, solve 0 ⫽ ⫺1.056785714t2 ⫹ 8.259285714t ⫹ 74.07142857 Solution We use the quadratic formula with a ⫽ ⫺1.056785714, b ⫽ 8.259285714, c ⫽ 74.07142857 Thus

⫺b ⫾ 1b2 ⫺ 4ac 2a ⫺8.259285714 ⫾ 1(8.259285714)2 ⫺ 4(⫺1.056785714)(74.07142857) ⫽ 2(⫺1.056785714) ⫺8.259285714 ⫾ 1381.3263106 ⫽ ⫺2.113571428 ⫺8.259285714 ⫾ 19.52757821 ⬇ ⫺2.113571428

t⫽

We are interested only in the positive solution, so t⫽

⫺8.259285714 ⫺ 19.52757821 ⬇ 13.14687393 ⫺2.113571428

Therefore, the trust fund balance is projected to reach zero slightly more than 13 years after 2000, during the year 2014.

2.1 Quadratic Equations ●

Graphing Utilities

147

We can use graphing utilities to determine (or approximate) the solutions of quadratic equations. The solutions can be found by using commands (such as ZERO or SOLVER) or programs. TRACE can be used to approximate where the graph of the function y ⫽ f(x) intersects the x-axis. This x-intercept is also the value of x that makes the function zero, so it is called a zero of the function. Because this value of x makes f(x) ⫽ 0, it is also a solution (or root) of the equation 0 ⫽ f(x). ■

● EXAMPLE 8 Solving with Technology Find the solutions of the equation 0 ⫽ ⫺7x2 ⫹ 16x ⫺ 4. Solution To solve the equation, we ﬁnd the x-intercepts of the graph of y ⫽ ⫺7x2 ⫹ 16x ⫺ 4. We graph this function and use TRACE to approximate an x-value that gives y ⫽ 0. Tracing to or evaluating at x ⫽ 2 gives y ⫽ 0, so x ⫽ 2 is an x-intercept. See Figure 2.1(a). 6

6

Y1=-7x2+16x-4

Y1=-7x2+16x-4

-2

3

X=2

-2

Zero X=.28571429

Y=0 -5

Figure 2.1

3

Y=0

-5

(a)

(b)

Tracing near x ⫽ 0.3 gives a value of y near 0, and an (approximate) x-intercept near 0.3 can be found more accurately by using ZERO. This gives y ⫽ 0 at x ⫽ 0.28571429 (approximately). See Figure 2.1(b). Solving f(x) ⫽ 0 algebraically gives solutions x ⫽ 2 and x ⫽ 72. 0 ⫽ (⫺7x ⫹ 2)(x ⫺ 2) ⫺7x ⫹ 2 ⫽ 0 or x ⫺ 2 ⫽ 0 2 x⫽ x⫽2 7 Spreadsheet Note

A spreadsheet such as Excel can also be used to solve a quadratic equation. However, because a quadratic equation f(x) ⫽ 0 may have two solutions, it is useful to graph the quadratic function y ⫽ f(x) before starting Goal Seek to solve the equation with Excel. For example, to solve the equation in Example 8, 0 ⫽ ⫺7x2 ⫹ 16x ⫺ 4, we use the graph of y ⫽ f(x) to approximate the x-intercepts and thus approximate the solutions to the equation. Graphing the equation shows that the x-intercepts are approximately 2 and 0 (see the graph in Figure 2.1). Entering x, the function, and values of x near 2 and 0 prepares us to use Goal Seek to ﬁnd the two solutions.

1 2 3

A x 2 0

B ⫺7x2 ⫹ 16x ⫺ 4 ⫽ ⫺7*A2^2 ⫹ 16*A2 ⫺ 4 ⫽ ⫺7*A3^2 ⫹ 16*A3 ⫺ 4

148

● Chapter 2

Quadratic and Other Special Functions

Entering 2 in A2 conﬁrms that x ⫽ 2 is a solution of the equation because B2 is 0, and using Goal Seek with B3 set to 0 gives the second solution as approximately 0.285688. This solution is an approximation of the exact solution x ⫽ 2兾7, accurate to four decimal places. A x 2 0.285688

1 2 3

●

●

Checkpoint

Checkpoint Solutions

B ⫹ 16x ⫺ 4 0 ⫺0.00032

⫺7x2

■

3. The statement of the quadratic formula says that if __________ and a ⫽ 0, then x ⫽ __________. 4. Solve 2x2 ⫺ 5x ⫽ 9 using the quadratic formula. 1. Zero; zero 2. (a) x2 ⫺ 19x ⫺ 20 ⫽ 0 (x ⫺ 20)(x ⫹ 1) ⫽ 0 x ⫺ 20 ⫽ 0 x⫹1⫽0 x ⫽ 20 x ⫽ ⫺1 (b) 2x2 ⫺ 6x ⫽ 0 2x(x ⫺ 3) ⫽ 0 2x ⫽ 0 x⫺3⫽0 x⫽0 x⫽3 3. If ax2 ⫹ bx ⫹ c ⫽ 0 with a ⫽ 0, then

⫺b ⫾ 1b2 ⫺ 4ac 2a 4. 2x2 ⫺ 5x ⫺ 9 ⫽ 0, so a ⫽ 2, b ⫽ ⫺5, c ⫽ ⫺9 x⫽

x⫽

x⫽

5 ⫾ 125 ⫺ 4(2)(⫺9) ⫺b ⫾ 1b2 ⫺ 4ac ⫽ 2a 4 5 ⫾ 125 ⫹ 72 ⫽ 4 5 ⫾ 197 ⫽ 4

5 ⫹ 197 ⬇ 3.712 4

or

x⫽

5 ⫺ 197 ⬇ ⫺1.212 4

2.1 Exercises In Problems 1–4, write the equations in general form. 1. 2x2 ⫹ 3 ⫽ x2 ⫺ 2x ⫹ 4 2. x2 ⫺ 2x ⫹ 5 ⫽ 2 ⫺ 2x2 3. (y ⫹ 1)(y ⫹ 2) ⫽ 4 4. (z ⫺ 1)(z ⫺ 3) ⫽ 1

In Problems 5–10, solve each equation by factoring. 5. 9 ⫺ 4x2 ⫽ 0 6. 25x2 ⫺ 16 ⫽ 0 2 7. x ⫽ x 8. t2 ⫺ 4t ⫽ 3t2 2 9. 4t ⫺ 4t ⫹ 1 ⫽ 0 10. 49z2 ⫹ 14z ⫹ 1 ⫽ 0

2.1 Quadratic Equations ●

In Problems 11–14, solve each equation by using the quadratic formula. Give real answers (a) exactly and (b) rounded to two decimal places. 11. x2 ⫺ 4x ⫺ 4 ⫽ 0 12. x2 ⫺ 6x ⫹ 7 ⫽ 0 13. 2w2 ⫹ w ⫹ 1 ⫽ 0 14. z2 ⫹ 2z ⫹ 4 ⫽ 0 In Problems 15–20, ﬁnd the exact real solutions to each equation, if they exist. 15. y2 ⫽ 7 16. z2 ⫽ 12 2 17. 5x ⫽ 80 18. 3x2 ⫽ 75 2 19. (x ⫹ 4) ⫽ 25 20. (x ⫹ 1)2 ⫽ 2 In Problems 21–30, use any method to ﬁnd the exact real solutions, if they exist. 21. x2 ⫹ 5x ⫽ 21 ⫹ x 22. x2 ⫹ 17x ⫽ 8x ⫺ 14 y2 w2 w 11 23. 24. ⫺ ⫺4⫽0 ⫺ y⫹1⫽0 8 2 2 6 25. 16z2 ⫹ 16z ⫺ 21 ⫽ 0 26. 10y2 ⫺ y ⫺ 65 ⫽ 0 27. (x ⫺ 1)(x ⫹ 5) ⫽ 7 28. (x ⫺ 3)(1 ⫺ x) ⫽ 1 29. 5x2 ⫽ 2x ⫹ 6 30. 3x2 ⫽ ⫺6x ⫺ 2 In Problems 31–36, solve each equation by using a graphing utility. 31. 21x ⫹ 70 ⫺ 7x2 ⫽ 0 32. 3x2 ⫺ 11x ⫹ 6 ⫽ 0 33. 300 ⫺ 2x ⫺ 0.01x2 ⫽ 0 34. ⫺9.6 ⫹ 2x ⫺ 0.1x2 ⫽ 0 35. 25.6x2 ⫺ 16.1x ⫺ 1.1 ⫽ 0 36. 6.8z2 ⫺ 4.9z ⫺ 2.6 ⫽ 0 In Problems 37–40, multiply both sides of the equation by the LCD, and solve the resulting quadratic equation. 8 x 3 37. x ⫹ ⫽ 9 38. ⫺1⫽ x x⫺2 x⫹1 x 1 5 3 39. 40. ⫽ 2x ⫹ ⫺ ⫽4 x⫺1 x⫺1 z⫹4 z⫺2 In Problems 41 and 42, solve using quadratic methods. 41. (x ⫹ 8)2 ⫹ 3(x ⫹ 8) ⫹ 2 ⫽ 0 42. (s ⫺ 2)2 ⫺ 5(s ⫺ 2) ⫺ 24 ⫽ 0 A P P L I C AT I O N S 43. Proﬁt If the proﬁt from the sale of x units of a product is P ⫽ 90x ⫺ 200 ⫺ x2, what level(s) of production will yield a proﬁt of $1200? 44. Proﬁt If the proﬁt from the sale of x units of a product is P ⫽ 16x ⫺ 0.1x2 ⫺ 100, what level(s) of production will yield a proﬁt of $180? 45. Proﬁt Suppose the proﬁt from the sale of x units of a product is P ⫽ 6400x ⫺ 18x2 ⫺ 400. (a) What level(s) of production will yield a proﬁt of $61,800? (b) Can a proﬁt of more than $61,800 be made? 46. Proﬁt Suppose the proﬁt from the sale of x units of a product is P ⫽ 50x ⫺ 300 ⫺ 0.01x2.

149

(a) What level(s) of production will yield a proﬁt of $250? (b) Can a proﬁt of more than $250 be made? 47. Flight of a ball If a ball is thrown upward at 96 feet per second from the top of a building that is 100 feet high, the height of the ball can be modeled by S ⫽ 100 ⫹ 96t ⫺ 16t2 feet where t is the number of seconds after the ball is thrown. How long after it is thrown is the height 100 feet? 48. Flight of a ball A tennis ball is thrown into the air from the top of a hotel that is 350 feet above the ground. The height of the ball from the ground is modeled by D(t) ⫽ ⫺16t2 ⫹ 10t ⫹ 350 feet where t is the time, in seconds, after the ball is thrown. How long after the ball is thrown does it hit the ground? 49. Wind and pollution The amount of airborne particulate pollution p from a power plant depends on the wind speed s, among other things, with the relationship between p and s approximated by p ⫽ 25 ⫺ 0.01s2 (a) Find the value(s) of s that will make p ⫽ 0. (b) What value of s makes sense in the context of this application? What does p ⫽ 0 mean in this application? 50. Drug sensitivity The sensitivity S to a drug is related to the dosage size by S ⫽ 100x ⫺ x2 where x is the dosage size in milliliters. (a) What dosage(s) will yield 0 sensitivity? (b) Explain what your answer in part (a) might mean. 51. Corvette acceleration The time t, in seconds, that it takes a 2008 Corvette to accelerate to x mph can be described by t ⫽ 0.001(0.732x2 ⫹ 15.417x ⫹ 607.738) (Source: Motor Trend). How fast is the Corvette going after 8.99 seconds? Give your answer to the nearest tenth. 52. Social Security trust fund Social Security beneﬁts are paid from a trust fund. As mentioned in the Application Preview, the trust fund balance, B, in billions of dollars, t years past the year 2000 is projected to be described by B ⫽ ⫺1.056785714t2 ⫹ 8.259285714t ⫹ 74.07142857 (Source: Social Security Administration projections). Find in what year the trust fund balance is projected to be $1000 billion in the red—that is, when B ⫽ ⫺1000.

150

● Chapter 2

Quadratic and Other Special Functions

53. Personal taxes Suppose that the percent of total personal income that is used to pay personal taxes is given by

58. Tourism spending The global spending on travel and tourism (in billions of dollars) can be described by the equation

y ⫽ 0.034x2 ⫺ 0.044x ⫹ 12.642

y ⫽ 6.75x2 ⫺ 122.94x ⫹ 1009.44

where x is the number of years past 1990 (Source: Bureau of Economic Analysis, U.S. Department of Commerce). (a) Verify that x ⫽ 7 is a solution of 14 ⫽ 0.034x2 ⫺ 0.044x ⫹ 12.642. (b) Find the year or years when the percent of total personal income used to pay personal taxes is 14.

where x equals the number of years past 1990 (Source: World Tourism Organization). Find the year after 1990 in which spending is projected to reach $983.52 billion.

54. Depth of a ﬁssure A ﬁssure in the earth appeared after an earthquake. To measure its vertical depth, a stone was dropped into it, and the sound of the stone’s impact was heard 3.9 seconds later. The distance (in feet) the stone fell is given by s ⫽ 16t12, and the distance (in feet) the sound traveled is given by s ⫽ 1090t2. In these equations, the distances traveled by the sound and the stone are the same, but their times are not. Using the fact that the total time is 3.9 seconds, ﬁnd the depth of the ﬁssure. 55. Marijuana use For the years from 1995 to 2006, the percent p of high school seniors who have tried marijuana can be considered as a function of time t according to p ⫽ f(t) ⫽ ⫺0.22t2 ⫹ 4.49t ⫹ 26.3 where t is the number of years past 1990 (Source: National Institute on Drug Abuse). In what year after 1990 will the percent predicted by the function fall to 18.6% if this function remains valid? 56. Projectile motion Two projectiles are shot into the air over a lake. The paths of the projectiles are given by (a) y ⫽ ⫺0.0013x2 ⫹ x ⫹ 10 and x2 4 (b) y ⫽ ⫺ ⫹ x ⫹ 10 81 3 where y is the height and x is the horizontal distance traveled. Determine which projectile travels farther by substituting y ⫽ 0 in each equation and ﬁnding x. 57. Percent proﬁt The Ace Jewelry Store sold a necklace for $144. If the percent proﬁt (based on cost) equals the cost of the necklace to the store, how much did the store pay for it? Use P⫽a

C bⴢC 100

where P is proﬁt and C is cost.

59. Internet use An equation that models the number of U.S. users of the Internet is y ⫽ ⫺0.261x2 ⫹ 21.748x ⫺ 82.503 million users where x is the number of years since 1990 (Source: Mediamark Research, New York). If the pattern indicated by the model remains valid, when does this model predict there will be 259.1 million users? 60. Velocity of blood Because of friction from the walls of an artery, the velocity of a blood corpuscle in an artery is greatest at the center of the artery and decreases as the distance r from the center increases. The velocity of the blood in the artery can be modeled by the function v ⫽ k(R2 ⫺ r2) where R is the radius of the artery and k is a constant that is determined by the pressure, the viscosity of the blood, and the length of the artery. In the case where k ⫽ 2 and R ⫽ 0.1 centimeters, the velocity is v ⫽ 2(0.01 ⫺ r2) centimeters per second. (a) What distance r would give a velocity of 0.02 cm/sec? (b) What distance r would give a velocity of 0.015 cm/sec? (c) What distance r would give a velocity of 0 cm/sec? Where is the blood corpuscle? 61. Body-heat loss The model for body-heat loss depends on the coefﬁcient of convection K, which depends on wind speed v according to the equation K 2 ⫽ 16v ⫹ 4 where v is in miles per hour. Find the positive coefﬁcient of convection when the wind speed is: (a) 20 mph (b) 60 mph (c) What is the change in K for a change in speed from 20 mph to 60 mph?

2.2 Quadratic Functions: Parabolas ●

2.2

151

Quadratic Functions: Parabolas

OBJECTIVES ●

●

●

●

To find the vertex of the graph of a quadratic function To determine whether a vertex is a maximum point or a minimum point To find the zeros of a quadratic function To graph quadratic functions

] Application Preview Because additional equipment, raw materials, and labor may cause variable costs of some products to increase dramatically as more units are produced, total cost functions are not always linear functions. For example, suppose that the total cost of producing a product is given by the equation C ⴝ C(x) ⴝ 300x ⴙ 0.1x2 ⴙ 1200 where x represents the number of units produced. That is, the cost of this product is represented by a second-degree function, or a quadratic function. We can find the cost of producing 10 units by evaluating C(10) ⴝ 300(10) ⴙ 0.1(10)2 ⴙ 1200 ⴝ 4210 If the revenue function for this product is R ⴝ R(x) ⴝ 600x then the profit is also a quadratic function: P ⴝ R ⴚ C ⴝ 600x ⴚ (300x ⴙ 0.1x2 ⴙ 1200) P ⴝ ⴚ0.1x2 ⴙ 300x ⴚ 1200 We can use this function to find how many units give maximum profit and what that profit is. (See Example 2.) In this section we will describe ways to find the maximum point or minimum point for a quadratic function.

Parabolas

In Chapter 1, we studied functions of the form y ⫽ ax ⫹ b, called linear (or ﬁrst-degree) functions. We now turn our attention to quadratic (or second-degree) functions. The general equation of a quadratic function has the form y ⫽ f(x) ⫽ ax2 ⫹ bx ⫹ c where a, b, and c are real numbers and a ⫽ 0. The graph of a quadratic function, y ⫽ ax2 ⫹ bx ⫹ c

(a ⫽ 0)

has a distinctive shape called a parabola. The basic function y ⫽ x2 and a variation of it, y ⫽ ⫺21x2, are parabolas whose graphs are shown in Figure 2.2. y

y

Vertex (maximum)

Axis of 4 symmetry

x

y = x2

-2

2

2 -2 x −2

2

Vertex (minimum)

Figure 2.2

1

y = − 2 x2 -4

(a)

(b)

Axis of symmetry

152

● Chapter 2

Quadratic and Other Special Functions

Vertex of a Parabola

As these examples illustrate, the graph of y ⫽ ax2 is a parabola that opens upward if a ⬎ 0 and downward if a ⬍ 0. The vertex, where the parabola turns, is a minimum point if a ⬎ 0 and a maximum point if a ⬍ 0. The vertical line through the vertex of a parabola is called the axis of symmetry because one half of the graph is a reﬂection of the other half through this line. The graph of y ⫽ (x ⫺ 2)2 ⫺ 1 is the graph of y ⫽ x2 shifted to a new location that is 2 units to the right and 1 unit down; its vertex is shifted from (0, 0) to (2, ⫺1) and its axis of symmetry is shifted 2 units to the right. (See Figure 2.3(a).) The graph of y ⫽ ⫺12(x ⫹ 1)2 ⫹ 2 is the graph of y ⫽ ⫺21x2 shifted 1 unit to the left and 2 units upward, with its vertex at (⫺1, 2). (See Figure 2.3(b).) y 3

(−1, 2)

1

y = − 2 (x + 1) 2 + 2 1 x −4

−3

−2

−1

1

2

−1 −2 −3

(a)

Figure 2.3

(b)

We can ﬁnd the x-coordinate of the vertex of the graph of y ⫽ ax2 ⫹ bx ⫹ c by using the fact that the axis of symmetry of a parabola passes through the vertex. Regardless of the location of the vertex of y ⫽ ax2 ⫹ bx ⫹ c or the direction it opens, the y-intercept of the graph of y ⫽ ax2 ⫹ bx ⫹ c is (0, c) and there is another point on the graph with y-coordinate c. See Figure 2.4. y

c

x

Figure 2.4

The x-coordinates of the points on this graph with y-coordinate c satisfy c ⫽ ax2 ⫹ bx ⫹ c Solving this equation gives 0 ⫽ ax2 ⫹ bx 0 ⫽ x(ax ⫹ b) x⫽0

or x ⫽

⫺b a

153

2.2 Quadratic Functions: Parabolas ●

The x-coordinate of the vertex is on the axis of symmetry, which is halfway from x ⫽ 0 to ⫺b ⫺b , at x ⫽ x⫽ . Thus we have the following. a 2a Vertex of a Parabola

The quadratic function y ⫽ f(x) ⫽ ax2 ⫹ bx ⫹ c has its vertex at a

⫺b , 2a

fa

⫺b bb 2a

The optimum value (either maximum or minimum) of the function occurs at x ⫽ and is f a

⫺b b. See Figure 2.5. 2a Axis of symmetry

y

⫺b 2a

y

Vertex (maximum)

y = ax 2 + bx + c (a > 0) −b 2a

f −b 2a

( )

Axis of symmetry x

x

f −b 2a

( )

Figure 2.5

Vertex (minimum)

−b 2a

y = ax 2 + bx + c (a < 0)

(b)

(a)

If we know the location of the vertex and the direction in which the parabola opens, we need very few other points to make a good sketch.

● EXAMPLE 1 Vertex and Graph of a Parabola Find the vertex and sketch the graph of y

f(x) ⫽ 2x2 ⫺ 4x ⫹ 4 8

Solution Because a ⫽ 2 ⬎ 0, the graph of f(x) opens upward and the vertex is the minimum point. We can calculate its coordinates as follows: ⫺(⫺4) ⫺b ⫽ ⫽1 2a 2(2) y ⫽ f (1) ⫽ 2

6

x⫽

Thus the vertex is (1, 2). Using x-values on either side of the vertex to plot additional points enables us to sketch the graph accurately. (See Figure 2.6.)

(2, 4)

4 2

y = 2x 2 − 4x + 4 (1, 2) x

−4

−2

2 −2

Figure 2.6

4

6

154

● Chapter 2

Quadratic and Other Special Functions

We can also use the coordinates of the vertex to ﬁnd maximum or minimum values without a graph. ] EXAMPLE 2 Profit Maximization (Application Preview) For the proﬁt function P(x) ⫽ ⫺0.1x2 ⫹ 300x ⫺ 1200 ﬁnd the number of units that give maximum proﬁt and ﬁnd the maximum proﬁt. Solution P(x) is a quadratic function with a ⬍ 0. Thus the graph of y ⫽ P(x) is a parabola that opens downward, so the vertex is a maximum point. The coordinates of the vertex are ⫺b ⫺300 ⫽ ⫽ 1500 2a 2(⫺0.1) P ⫽ P(1500) ⫽ ⫺0.1(1500)2 ⫹ 300(1500) ⫺ 1200 ⫽ 223,800 x⫽

Therefore, the maximum proﬁt is $223,800 when 1500 units are sold. Graphing Utilities

We can also use a graphing utility to graph a quadratic function. Even with a graphing tool, recognizing that the graph is a parabola and locating the vertex are important. They help us to set the viewing window (to include the vertex) and to know when we have a complete graph. For example, suppose we wanted to graph the proﬁt function from Example 2 (and the Application Preview). Because x ⱖ 0 and the vertex and axis of symmetry are at x ⫽ 1500, we can set the x-range so that it includes x ⫽ 0 and has x ⫽ 1500 near its center. For the P-range (or y-range), we know that the maximum is P ⫽ 223,800 and that P(0) ⫽ ⫺1200, so the window should include these y-values. Figure 2.7 shows the graph. Note that the graph of this function has two x-intercepts. 240,000

0 -1500

3000

Figure 2.7

Zeros of Quadratic Functions

■

As we noted in Chapter 1, “Linear Equations and Functions,” the x-intercepts of the graph of a function y ⫽ f(x) are the values of x for which f(x) ⫽ 0, called the zeros of the function. As we saw in the previous section, the zeros of the quadratic function y ⫽ f(x) ⫽ ax2 ⫹ bx ⫹ c are the solutions of the quadratic equation ax2 ⫹ bx ⫹ c ⫽ 0 ⫺b ⫾ 1b2 ⫺ 4ac 2a

Therefore, the zeros are given by the quadratic formula as x⫽

2.2 Quadratic Functions: Parabolas ●

155

The information that is useful in graphing quadratic functions is summarized as follows. Graphs of Quadratic Functions

Form: y ⫽ f(x) ⫽ ax2 ⫹ bx ⫹ c Graph: parabola a ⬎ 0 parabola opens upward; vertex is a minimum point a ⬍ 0 parabola opens downward; vertex is a maximum point Coordinates of vertex: x ⫽

⫺b , 2a

Axis of symmetry equation: x ⫽

y⫽fa

⫺b 2a

⫺b b 2a y

⫺b ⫹ 1b2 ⫺ 4ac x⫽ , 2a ⫺b ⫺ 1b2 ⫺ 4ac x⫽ 2a

x-intercepts or zeros (if real*):

Axis of symmetry

y Vertex (maximum) y = ax2 + bx + c (a > 0)

−b 2a

x f −b 2a

( )

Vertex (minimum)

f −b 2a

( )

Axis of symmetry

x −b 2a

y = ax2 + bx + c (a < 0)

y-intercept: Let x ⫽ 0; then y ⫽ c.

● EXAMPLE 3 Graph of a Quadratic Function For the function y ⫽ 4x ⫺ x2, determine whether its vertex is a maximum point or a minimum point and ﬁnd the coordinates of this point, ﬁnd the zeros, if any exist, and sketch the graph. Solution The proper form is y ⫽ ⫺x2 ⫹ 4x ⫹ 0, so a ⫽ ⫺1. Thus the parabola opens downward, and the vertex is the highest (maximum) point. ⫺b ⫺4 ⫽ ⫽ 2. 2a 2(⫺1) The y-coordinate of the vertex is f(2) ⫽ ⫺(2)2 ⫹ 4(2) ⫽ 4. The zeros of the function are solutions to The vertex occurs at x ⫽

⫺x2 ⫹ 4x ⫽ 0 x(⫺x ⫹ 4) ⫽ 0 or x ⫽ 0 and x ⫽ 4 The graph of the function can be found by drawing a parabola with these three points (see Figure 2.8(a)), or by using a graphing utility or a spreadsheet (see Figure 2.8(b)). y

Vertex (2, 4) maximum

4

6 4 2

y

2

y = 4x − x 2

-2 x

-1

1

2

3

0

2

-2

4

-4

x Figure 2.8

(a) *If the zeros are not real, the graph does not cross the x-axis.

(b)

4

6

156

● Chapter 2

Quadratic and Other Special Functions

Example 3 shows how to use the solutions of f(x) ⫽ 0 to ﬁnd the x-intercepts of a parabola if y ⫽ f(x) is a quadratic function. Conversely, we can use the x-intercepts of y ⫽ f(x), if they exist, to ﬁnd or approximate the solutions of f(x) ⫽ 0. Exact integer solutions found in this way can also be used to determine factors of f(x).

● EXAMPLE 4 Graph Comparison Figure 2.9 shows the graphs of two different quadratic functions. Use the ﬁgure to answer the following. (a) Determine the vertex of each function. (b) Determine the real solutions of f1(x) ⫽ 0 and f2(x) ⫽ 0. (c) One of the graphs in Figure 2.9 is the graph of y ⫽ 7 ⫹ 6x ⫺ x2, and one is the graph of y ⫽ x2 ⫺ 6x ⫹ 10. Determine which is which, and why. Solution (a) For y ⫽ f1(x), the vertex is the maximum point, at (3, 16). For y ⫽ f2(x), the vertex is the minimum point, at (3, 1). (b) For y ⫽ f1(x), the real solutions of f1(x) ⫽ 0 are the zeros, or x-intercepts, at x ⫽ ⫺1 and x ⫽ 7. For y ⫽ f2(x), the graph has no x-intercepts, so f2(x) ⫽ 0 has no real solutions. (c) Because the graph of y ⫽ f1(x) opens downward, the coefﬁcient of x2 must be negative. Hence Figure 2.9(a) shows the graph of y ⫽ f1(x) ⫽ 7 ⫹ 6x ⫺ x2. Similarly, the coefﬁcient of x2 in y ⫽ f2(x) must be positive, so Figure 2.9(b) shows the graph of y ⫽ f2(x) ⫽ x2 ⫺ 6x ⫹ 10. y

y

15

y = f1(x) 10

10

5

5

y = f2 (x) x

x -4

Figure 2.9

●

Checkpoint

-2

2

4

6

8

10

(a)

-4

-2

2

4

6

8

10

(b)

1. Name the graph of a quadratic function. 2. (a) What is the x-coordinate of the vertex of y ⫽ ax2 ⫹ bx ⫹ c? (b) For y ⫽ 12x ⫺ 21x2, what is the x-coordinate of the vertex? What is the y-coordinate of the vertex? 3. (a) How can you tell whether the vertex of f(x) ⫽ ax2 ⫹ bx ⫹ c is a maximum point or a minimum point? (b) In 2(b), is the vertex a maximum point or a minimum point? 4. The zeros of a function correspond to what feature of its graph?

● EXAMPLE 5 Maximizing Revenue Ace Cruises offers a sunset cruise to a group of 50 people for a price of $30 per person, but it reduces the price per person by $0.50 for each additional person above the 50.

2.2 Quadratic Functions: Parabolas ●

157

(a) Does reducing the price per person to get more people in the group give the cruise company more revenue? (b) How many people will provide maximum revenue for the cruise company? Solution (a) The revenue to the company if 50 people are in the group and each pays $30 is 50($30) ⫽ $1500. The following table shows the revenue for the addition of people to the group. The table shows that as the group size begins to increase past 50 people, the revenue also increases. However, it also shows that increasing the group size too much (to 70 people) reduces revenue. Increase in Group Size

Number of People

Decrease in Price

New Price ($)

Revenue ($)

0 1 2 3 o 20 o x

50 51 52 53 o 70 o 50 ⫹ x

0 0.50(1) 0.50(2) ⫽ 1 0.50(3) ⫽ 1.50 o 0.50(20) ⫽ 10 o 0.50(x)

30 29.50 29 28.50 o 20 o 30 ⫺ 0.50x

50(30) ⫽ 1500 51(29.50) ⫽ 1504.50 52(29) ⫽ 1508 53(28.50) ⫽ 1510.50 o 70(20) ⫽ 1400 o (50 ⫹ x)(30 ⫺ 0.50x)

(b) The last entry in the table shows the revenue for an increase of x people in the group. R(x) ⫽ (50 ⫹ x)(30 ⫺ 0.50x) Expanding this function gives a form from which we can ﬁnd the vertex of its graph. R(x) ⫽ 1500 ⫹ 5x ⫺ 0.50x2 ⫺b ⫺5 ⫽ ⫽ 5, R(5) ⫽ 1512.50. 2a 2(⫺0.50) This means that the revenue will be maximized at $1512.50 when 50 ⫹ 5 ⫽ 55 people are in the group, with each paying 30 ⫺ 0.50(5) ⫽ 27.50 dollars. The vertex of the graph of this function is x ⫽

Average Rate of Change

Average Rate of Change

Because the graph of a quadratic function is not a line, the rate of change of the function is not constant. We can, however, ﬁnd the average rate of change of a function between two input values if we know how much the function output values change between the two input values. The average rate of change of f(x) with respect to x over the interval from x ⫽ a to x ⫽ b (where a ⬍ b) is calculated as Average rate of change ⫽

change of f(x) f(b) ⫺ f(a) ⫽ corresponding change in x-values b⫺a

The average rate of change is also the slope of the segment (or secant line) joining the points (a, f(a)) and (b, f(b)).

f (x)

(b, f(b))

m=

f (b) – f (a) b–a

(a, f (a)) a

x

b

158

● Chapter 2

Quadratic and Other Special Functions

● EXAMPLE 6 Average Rate of Change of Revenue We found that the revenue for the cruise in Example 5 was deﬁned by the quadratic function R(x) ⫽ 1500 ⫹ 5x ⫺ 0.50x2 where x is the increase in the group size beyond 50 people. What is the average rate of change of revenue if the group increases from 50 to 55 persons? Solution The average rate of change of revenue from 50 to 55 persons is the average rate of change of the function from x ⫽ 0 to x ⫽ 5. R(5) ⫺ R(0) 1512.50 ⫺ 1500 ⫽ ⫽ 2.50 dollars per person 5⫺0 5 This average rate of change is also the slope of the secant line connecting the points (0, 1500) and (5, 1512.50) on the graph of the function (see Figure 2.10). R(x)

R(x) = 1500 + 5x – 0.50x 2 (5, 1512.50) 1510

(0, 1500)

x

Figure 2.10

●

Checkpoint Solutions

m = 1512.50 – 1500 = 2.50 5–0

2

4

6

8

1. Parabola ⫺b 2a ⫺12 ⫺12 (b) x ⫽ ⫽ ⫽ 12 2(⫺1兾2) (⫺1) 1 y ⫽ 12(12) ⫺ (12)2 ⫽ 144 ⫺ 72 ⫽ 72 2 3. (a) Maximum point if a ⬍ 0; minimum point if a ⬎ 0. (b) (12, 72) is a maximum point because a ⫽ ⫺12. 4. The x-intercepts 2. (a) x ⫽

2.2 Exercises In Problems 1–6, (a) ﬁnd the vertex of the graph of the equation, (b) determine if the vertex is a maximum or minimum point, (c) determine what value of x gives the optimal value of the function, and (d) determine the optimal (maximum or minimum) value of the function. 1 1. y ⫽ x2 ⫹ x 2. y ⫽ x2 ⫺ 2x 2 3. y ⫽ 8 ⫹ 2x ⫺ x2 4. y ⫽ 6 ⫺ 4x ⫺ 2x2 2 5. f(x) ⫽ 6x ⫺ x 6. f(x) ⫽ x2 ⫹ 2x ⫺ 3

In Problems 7–12, determine whether each function’s vertex is a maximum point or a minimum point and ﬁnd the coordinates of this point. Find the zeros, if any exist, and sketch the graph of the function. 1 7. y ⫽ x ⫺ x2 8. y ⫽ ⫺2x2 ⫹ 18x 4 9. y ⫽ x2 ⫹ 4x ⫹ 4 10. y ⫽ x2 ⫺ 6x ⫹ 9 1 11. x2 ⫹ x ⫺ y ⫺ 3 ⫽ 0 12. x2 ⫹ x ⫹ 2y ⫽ 5 2

2.2 Quadratic Functions: Parabolas ●

159

For each function in Problems 13–16, (a) tell how the graph of y ⴝ x2 is shifted, and (b) graph the function. 13. y ⫽ (x ⫺ 3)2 ⫹ 1 14. y ⫽ (x ⫺ 10)2 ⫹ 1 15. y ⫽ (x ⫹ 2)2 ⫺ 2 16. y ⫽ (x ⫹ 12)2 ⫺ 8

33. Crop yield The yield in bushels from a grove of orange trees is given by Y ⫽ x(800 ⫺ x), where x is the number of orange trees per acre. How many trees will maximize the yield?

In Problems 17–20, graph each function with a graphing utility. Use the graph to ﬁnd the vertex and zeros. Check your results algebraically. 1 15 17. y ⫽ x2 ⫺ x ⫺ 2 2 18. y ⫽ 0.1(x2 ⫹ 4x ⫺ 32) 1 19. y ⫽ x2 ⫹ 3x ⫹ 12 4 20. y ⫽ x2 ⫺ 2x ⫹ 5

34. Stimulus-response One of the early results in psychology relating the magnitude of a stimulus x to the magnitude of a response y is expressed by the equation

In Problems 21–22, ﬁnd the average rate of change of the function between the given values of x. 21. y ⫽ ⫺5x ⫺ x2 between x ⫽ ⫺1 and x ⫽ 1. 22. y ⫽ 8 ⫹ 3x ⫹ 0.5x2 between x ⫽ 2 and x ⫽ 4. In Problems 23–26, ﬁnd the vertex and zeros and use them to determine a range for a graphing utility that includes these values; graph the function with that range. 23. y ⫽ 63 ⫹ 0.2x ⫺ 0.01x2 24. y ⫽ 0.2x2 ⫹ 16x ⫹ 140 25. y ⫽ 0.0001x2 ⫺ 0.01 26. y ⫽ 0.01x ⫺ 0.001x2 In Problems 27 and 28, (a) ﬁnd the vertex of each function f(x). Use the vertex to set the window in which to graph y ⴝ f(x) and then (b) graphically approximate the solutions to f(x) ⴝ 0. 27. f(x) ⫽ 8x2 ⫺ 16x ⫺ 16 28. f(x) ⫽ 3x2 ⫺ 18x ⫹ 16 In Problems 29 and 30, complete the following. (a) Use the graph of y ⴝ f(x) to ﬁnd an integer solution to f(x) ⴝ 0. (b) Use the solution from (a) to ﬁnd a factor of f(x). (c) Factor f(x). (d) Solve f(x) ⴝ 0. 29. f(x) ⫽ 3x2 ⫺ 8x ⫹ 4 30. f(x) ⫽ 5x2 ⫺ 2x ⫺ 7 A P P L I C AT I O N S 31. Proﬁt The daily proﬁt from the sale of a product is given by P ⫽ 16x ⫺ 0.1x2 ⫺ 100 dollars. (a) What level of production maximizes proﬁt? (b) What is the maximum possible proﬁt? 32. Proﬁt The daily proﬁt from the sale of x units of a product is P ⫽ 80x ⫺ 0.4x2 ⫺ 200 dollars. (a) What level of production maximizes proﬁt? (b) What is the maximum possible proﬁt?

y ⫽ kx2 where k is an experimental constant. Sketch this graph for k ⫽ 1, k ⫽ 2, and k ⫽ 4. 35. Drug sensitivity The sensitivity S to a drug is related to the dosage x (in milligrams) by S ⫽ 1000x ⫺ x2 Sketch the graph of this function and determine what dosage gives maximum sensitivity. Use the graph to determine the maximum sensitivity. 36. Maximizing an enclosed area If 100 feet of fence is used to enclose a rectangular yard, then the resulting area is given by A ⫽ x(50 ⫺ x) where x feet is the width of the rectangle and 50 ⫺ x feet is the length. Graph this equation and determine the length and width that give maximum area. 37. Photosynthesis The rate of photosynthesis R for a certain plant depends on the intensity of light x, in lumens, according to R ⫽ 270x ⫺ 90x2 Sketch the graph of this function, and determine the intensity that gives the maximum rate. 38. Projectiles A ball thrown vertically into the air has its height above ground given by s ⫽ 112t ⫺ 16t2 where t is in seconds and s is in feet. Find the maximum height of the ball. 39. Projectiles Two projectiles are shot into the air from the same location. The paths of the projectiles are parabolas and are given by (a) y ⫽ ⫺0.0013x2 ⫹ x ⫹ 10 and ⫺x2 4 (b) y ⫽ ⫹ x ⫹ 10 81 3 where x is the horizontal distance and y is the vertical distance, both in feet. Determine which projectile goes higher by locating the vertex of each parabola.

160

● Chapter 2

Quadratic and Other Special Functions

40. Flow rates of water The speed at which water travels in a pipe can be measured by directing the ﬂow through an elbow and measuring the height to which it spurts out the top. If the elbow height is 10 cm, the equation relating the height h (in centimeters) of the water above the elbow and its velocity v (in centimeters per second) is given by v2 ⫽ 1960(h ⫹ 10) Solve this equation for h and graph the result, using the velocity as the independent variable. 41. Cost The ﬁgure below shows the graph of a total cost function, with x equal to the number of units produced. (a) Is the average rate of change of cost greater from x ⫽ a to x ⫽ b or from x ⫽ b to x ⫽ c? Explain. (b) Would the number of units d need to satisfy d ⬍ b or d ⬎ b for the average rate of change of cost from x ⫽ a to x ⫽ d to be greater than that from x ⫽ a to x ⫽ b? Explain. C(x)

x

a

b

c

42. Revenue The ﬁgure below shows the graph of a total revenue function, with x equal to the number of units sold. (a) Is the average rate of change of revenue negative from x ⫽ a to x ⫽ b or from x ⫽ b to x ⫽ c? Explain. (b) Would the number of units d need to satisfy d ⬍ b or d ⬎ b for the average rate of change of revenue from x ⫽ a to x ⫽ d to be greater than that from x ⫽ a to x ⫽ b? Explain. R(x)

month, but she rents one fewer apartment for each $20 increase in monthly rent. (a) Construct a table that gives the revenue generated if she charges $600, $620, and $640. (b) Does her revenue from the rental of the apartments increase or decrease as she increases the rent from $600 to $640? (c) Write an equation that gives the revenue from rental of the apartments if she makes x increases of $20 in the rent. (d) Find the rent she should charge to maximize her revenue. 44. Revenue The owner of a skating rink rents the rink for parties at $600 if 50 or fewer skaters attend, so that the cost per person is $12 if 50 attend. For each 5 skaters above 50, she reduces the price per skater by $0.50. (a) Construct a table that gives the revenue generated if 50, 60, and 70 skaters attend. (b) Does the owner’s revenue from the rental of the rink increase or decrease as the number of skaters increases from 50 to 70? (c) Write the equation that describes the revenue for parties with x more than 50 skaters. (d) Find the number of skaters that will maximize the revenue. 45. Pension resources The Pension Beneﬁt Guaranty Corporation is the agency that insures pensions. The ﬁgure shows one study’s projection for the agency’s total resources, initially rising (from taking over the assets of failing plans) but then falling (as more workers retire and payouts increase). (a) What kind of function might be used to model the agency’s total resources? (b) If a function of the form f(x) ⫽ ax2 ⫹ bx ⫹ c were used to model these total resources, would f(x) have a ⬎ 0 or a ⬍ 0? Explain. (c) If the model from part (b) used x as the number of years past 2004, explain why the model would have b ⬎ 0 and c ⬎ 0. +$50 billion + 40 + 30 + 20 + 10

x

a

b

c

43. Apartment rental The owner of an apartment building can rent all 50 apartments if she charges $600 per

0 – 10 ’04

’08

’12

’16

’20

Source: The New York Times, September 14, 2004. Copyright © 2004 The New York Times Co. Reprinted by permission.

2.3 Business Applications of Quadratic Functions ●

46. Projectile motion When a stone is thrown upward, it follows a parabolic path given by a form of the equation y ⫽ ax2 ⫹ bx ⫹ c If y ⫽ 0 represents ground level, ﬁnd the equation of a stone that is thrown from ground level at x ⫽ 0 and lands on the ground 40 units away if the stone reaches a maximum height of 40 units. (Hint: Find the coordinates of the vertex of the parabola and two other points.) 47. Health care Many politicians are discussing national health insurance because health care costs are increasing so rapidly. Health care costs in the United States (in billions of dollars) are given by y ⫽ 5.033x2 ⫹ 100.5x ⫹ 1378 where x is the number of years past 2000 (Source: U.S. Centers for Medicare and Medicaid Services). Does the model indicate that the cost projections from 2010 to 2015 increase more rapidly than the costs during the years 2005 to 2010?

161

(a) Graph the function y ⫽ p(t). (b) From the graph, identify the maximum point on the graph of y ⫽ p(t). (c) In what year is the percent of women workers at its maximum, according to this model? Union participation The percent of U.S. workers who belonged to unions for selected years from 1930 to 2005 can be described by u(x) ⫽ ⫺0.013x2 ⫹ 1.56x ⫺ 18.87 where x is the number of years past 1900 (Source: World Almanac, 2007). Use this function in Problems 49–51. 49. Graph the function u(x). 50. For what year does the function u(x) indicate a maximum percentage of workers belonged to unions? 51. (a) For what years does the function u(x) predict that 0% of U.S. workers will belong to unions? (b) When can you guarantee that u(x) can no longer be used to describe the percent of U.S. workers who belong to unions?

48. Women in the work force The percent of the total work force that is female is given by p(t) ⫽ ⫺0.00934t2 ⫹ 0.852t ⫹ 21.98 where t is the number of years past 1960.

2.3

Business Applications of Quadratic Functions

OBJECTIVES ●

●

●

●

To graph quadratic supply and demand functions To find market equilibrium by using quadratic supply and demand functions To find break-even points by using quadratic cost and revenue functions To maximize quadratic revenue and profit functions

] Application Preview Suppose that the supply function for a product is given by the quadratic function p ⴝ q2 ⴙ 100 and the demand function is given by p ⴝ ⴚ20q ⴙ 2500. Finding the market equilibrium involves solving a quadratic equation (see Example 1). In this section, we graph quadratic supply and demand functions and find market equilibrium by solving supply and demand functions simultaneously using quadratic methods. We will also discuss quadratic revenue, cost, and profit functions, including break-even points and profit maximization.

Supply, Demand, and Market Equilibrium The ﬁrst-quadrant parts of parabolas or other quadratic equations are frequently used to represent supply and demand functions. For example, the ﬁrst-quadrant part of p ⫽ q2 ⫹ q ⫹ 2 (Figure 2.11(a) on the next page) may represent a supply curve, whereas the ﬁrst-quadrant part of q2 ⫹ 2q ⫹ 6p ⫺ 23 ⫽ 0 (Figure 2.11(b)) may represent a demand curve.

162

● Chapter 2

Quadratic and Other Special Functions p

p 10

10

q 2 + 2q + 6p − 23 = 0 8 (p ≥ 0, q ≥ 0)

8

6

6 4

p = q2 + q + 2 (p ≥ 0, q ≥ 0)

2 q

q −4

−2

2

−4

4

−2

2

4

−2

(a)

Figure 2.11

−2

(b)

When quadratic equations are used to represent supply or demand curves, we can solve their equations simultaneously to ﬁnd the market equilibrium as we did with linear supply and demand functions. As in Section 1.5, we can solve two equations in two variables by eliminating one variable and obtaining an equation in the other variable. When the functions are quadratic, the substitution method of solution is perhaps the best, and the resulting equation in one unknown will usually be quadratic. ] EXAMPLE 1 Supply and Demand (Application Preview) If the supply function for a commodity is given by p ⫽ q2 ⫹ 100 and the demand function is given by p ⫽ ⫺20q ⫹ 2500, ﬁnd the point of market equilibrium. Solution At market equilibrium, both equations will have the same p-value. Thus substituting q2 ⫹ 100 for p in p ⫽ ⫺20q ⫹ 2500 yields q2 ⫹ 100 ⫽ ⫺20q ⫹ 2500 ⫹ 20q ⫺ 2400 ⫽ 0 (q ⫺ 40)(q ⫹ 60) ⫽ 0 q ⫽ 40 or q ⫽ ⫺60

q2

Because a negative quantity has no meaning, the equilibrium point occurs when 40 units are sold, at (40, 1700). The graphs of the functions are shown (in the ﬁrst quadrant only) in Figure 2.12. p 2800 2500 2400

p = q 2 + 100

2000

(40, 1700)

1600 1200 800

p = −20q + 2500

400 100

Figure 2.12

q 10

20

30

40

50

60

70

2.3 Business Applications of Quadratic Functions ●

163

● EXAMPLE 2 Market Equilibrium If the demand function for a commodity is given by p(q ⫹ 4) ⫽ 400 and the supply function is given by 2p ⫺ q ⫺ 38 ⫽ 0, ﬁnd the market equilibrium. Solution Solving the supply equation 2p ⫺ q ⫺ 38 ⫽ 0 for p gives p ⫽ 21q ⫹ 19. Substituting for p in p(q ⫹ 4) ⫽ 400 gives 1 ¢ q ⫹ 19≤(q ⫹ 4) ⫽ 400 2 1 2 q ⫹ 21q ⫹ 76 ⫽ 400 2 1 2 q ⫹ 21q ⫺ 324 ⫽ 0 2 Multiplying both sides of the equation by 2 yields q2 ⫹ 42q ⫺ 648 ⫽ 0. Factoring gives (q ⫺ 12)(q ⫹ 54) ⫽ 0 q ⫽ 12 or q ⫽ ⫺54 Thus the market equilibrium occurs when 12 items are sold, at a price of p ⫽ 12(12) ⫹ 19 ⫽ $25 each. The graphs of the demand and supply functions are shown in Figure 2.13(a).

p 100

p(q + 4) = 400

80 60

2p − q − 38 = 0

40

(12, 25)

20 19 q 4

16

20

Graphing utilities also can be used to sketch these graphs. A command such as INTERSECT could be used to determine points of intersection that give market equilibrium. Figure 2.13(b) shows the graph of the supply and demand functions for the commodity in Example 2. Using the INTERSECT command gives the same market equilibrium point determined in Example 2 (see Figure 2.13(c)). 100

100

0

20 0

Figure 2.13 (continued)

12

(a)

Figure 2.13

Graphing Utilities

8

(b)

Intersection 0 X=12 0

(c)

Y=25

20

■

164

● Chapter 2 ●

Quadratic and Other Special Functions

Checkpoint

1. The point of intersection of the supply and demand functions is called __________. 2. If the demand and supply functions for a product are p⫹

1 2 1 q ⫽ 1000 and p ⫽ q ⫹ 10 10 10

respectively, ﬁnding the market equilibrium point requires solution of what equation? Find the market equilibrium.

Break-Even Points and Maximization In Chapter 1, “Linear Equations and Functions,” we discussed linear total cost and total revenue functions. Many total revenue functions may be linear, but costs tend to increase sharply after a certain level of production. Thus functions other than linear functions, including quadratic functions, are used to predict the total costs of products. For example, the monthly total cost curve for a commodity may be the parabola with equation C(x) ⫽ 360 ⫹ 40x ⫹ 0.1x2. If the total revenue function is R(x) ⫽ 60x, we can ﬁnd the break-even point by ﬁnding the quantity x that makes C(x) ⫽ R(x). (See Figure 2.14.) Setting C(x) ⫽ R(x), we have 360 ⫹ 40x ⫹ 0.1x2 ⫽ 60x 0.1x2 ⫺ 20x ⫹ 360 ⫽ 0 x2 ⫺ 200x ⫹ 3600 ⫽ 0 (x ⫺ 20)(x ⫺ 180) ⫽ 0 x ⫽ 20 or x ⫽ 180

$ 12,000 11,000 10,000 9000 8000 7000 6000 5000 4000 3000 2000 1000

C(x) = 360 + 40x + 0.1x 2

R(x) = 60x

Thus C(x) ⫽ R(x) at x ⫽ 20 and at x ⫽ 180. If 20 items are produced and sold, Break-even points C(x) and R(x) are both $1200; if 180 items are sold, C(x) and R(x) are both $10,800. Thus there are two break-even points. x In a monopoly market, the revenue of a 50 100 150 200 250 company is restricted by the demand for the product. In this case, the relationship between Figure 2.14 the price p of the product and the number of units sold x is described by the demand function p ⫽ f(x), and the total revenue function for the product is given by R ⫽ px ⫽ 3f(x)4x

If, for example, the demand for a product is given by p ⫽ 300 ⫺ x

where x is the number of units sold and p is the price, then the revenue function for this product is the quadratic function R ⫽ px ⫽ (300 ⫺ x)x ⫽ 300x ⫺ x2

● EXAMPLE 3 Break-Even Point Suppose that in a monopoly market the total cost per week of producing a high-tech product is given by C ⫽ 3600 ⫹ 100x ⫹ 2x2. Suppose further that the weekly demand function for this product is p ⫽ 500 ⫺ 2x. Find the number of units that will give the break-even point for the product.

2.3 Business Applications of Quadratic Functions ●

165

Solution The total cost function is C(x) ⫽ 3600 ⫹ 100x ⫹ 2x2, and the total revenue function is R(x) ⫽ (500 ⫺ 2x)x ⫽ 500x ⫺ 2x2. Setting C(x) ⫽ R(x) and solving for x gives 3600 ⫹ 100x ⫹ 2x2 ⫽ 500x ⫺ 2x2 4x2 ⫺ 400x ⫹ 3600 ⫽ 0 x2 ⫺ 100x ⫹ 900 ⫽ 0 (x ⫺ 90)(x ⫺ 10) ⫽ 0 x ⫽ 90 or x ⫽ 10 Does this mean the ﬁrm will break even at 10 units and at 90 units? Yes. Figure 2.15 shows the graphs of C(x) and R(x). From the graph we can observe that the ﬁrm makes a proﬁt after x ⫽ 10 until x ⫽ 90, because R(x) ⬎ C(x) in that interval. At x ⫽ 90, the proﬁt is 0, and the ﬁrm loses money if it produces more than 90 units per week. $ 40,000 36,000

Loss

32,000

(90, 28,800)

Max revenue (125, 31,250)

28,000 24,000

Pro

16,000 12,000

C(x) = 3600 + 100x + 2x 2

8000 4000

R(x) = 500x − 2x 2

fit

20,000

(10, 4800) x

Figure 2.15

●

Checkpoint

Loss

20

60

100

140

180

3. The point of intersection of the revenue function and the cost function is called ________. 4. If C(x) ⫽ 120x ⫹ 15,000 and R(x) ⫽ 370x ⫺ x2, ﬁnding the break-even points requires solution of what equation? Find the break-even points. Note that for Example 3, the revenue function R(x) ⫽ (500 ⫺ 2x)x ⫽ 500x ⫺ 2x2 is a parabola that opens downward. Thus the vertex is the point at which revenue is maximum. We can locate this vertex by using the methods discussed in the previous section. Vertex: x ⫽

⫺b ⫺500 500 ⫽ ⫽ ⫽ 125 (units) 2a 2(⫺2) 4

It is interesting to note that when x ⫽ 125, the ﬁrm achieves its maximum revenue of R(125) ⫽ 500(125) ⫺ 2(125)2 ⫽ 31,250 (dollars) but the costs when x ⫽ 125 are C(125) ⫽ 3600 ⫹ 100(125) ⫹ 2(125)2 ⫽ 47,350 (dollars) which results in a loss. This illustrates that maximizing revenue is not a good goal. We should seek to maximize proﬁt. Figure 2.15 shows that maximum proﬁt will occur where the distance between the revenue and cost curves (that is, R(x) ⫺ C(x)) is largest. This appears to be near x ⫽ 50, which is midway between the x-values of the break-even points. This is veriﬁed in the next example.

166

● Chapter 2

Quadratic and Other Special Functions

● EXAMPLE 4 Profit Maximization For the total cost function C(x) ⫽ 3600 ⫹ 100x ⫹ 2x2 and the total revenue function R(x) ⫽ 500x ⫺ 2x2 (from Example 3), ﬁnd the number of units that maximizes proﬁt and ﬁnd the maximum proﬁt. Solution Using Profit ⫽ Revenue ⫺ Cost, we can determine the proﬁt function: P(x) ⫽ (500x ⫺ 2x2) ⫺ (3600 ⫹ 100x ⫹ 2x2) ⫽ ⫺3600 ⫹ 400x ⫺ 4x2 This proﬁt function is a parabola that opens downward, so the vertex will be the maximum point. Vertex: x ⫽

⫺b ⫺400 ⫺400 ⫽ ⫽ ⫽ 50 2a 2(⫺4) ⫺8

Furthermore, when x ⫽ 50, we have P(50) ⫽ ⫺3600 ⫹ 400(50) ⫺ 4(50)2 ⫽ 6400 (dollars) Thus, when 50 items are produced and sold, a maximum proﬁt of $6400 is made (see Figure 2.16). P 8000

(50, 6400) Max profit

6400 4000

x −20

−4000

Figure 2.16

20 10

40 50 60

80 100 90

P(x) = −3600 + 400x − 4x 2

Figure 2.16 shows that the break-even points are at x ⫽ 10 and x ⫽ 90 and that the maximum proﬁt occurs at the x-value midway between these x-values. This is reasonable because the graph of the proﬁt function is a parabola, and the x-value of any parabola’s vertex occurs midway between its x-intercepts. It is important to note that the procedures for ﬁnding maximum revenue and proﬁt in these examples depend on the fact that these functions are parabolas. Using methods discussed in Sections 2.1 and 2.2, we can use graphing utilities to locate maximum points, minimum points, and break-even points. For more general functions, procedures for ﬁnding maximum or minimum values are discussed in Chapter 10, “Applications of Derivatives.” ●

Checkpoint Solutions

1. The market equilibrium point or market equilibrium 1 1 2. Demand: p ⫽ ⫺ q2 ⫹ 1000; supply: p ⫽ q ⫹ 10 10 10 1 1 Solution of ⫺ q2 ⫹ 1000 ⫽ q ⫹ 10 10 10 ⫺q2 ⫹ 10,000 ⫽ q ⫹ 100 0 ⫽ q2 ⫹ q ⫺ 9900 0 ⫽ (q ⫹ 100)(q ⫺ 99) q ⫽ ⫺100 or q ⫽ 99 Thus market equilibrium occurs when q ⫽ 99 and p ⫽ 9.9 ⫹ 10 ⫽ 19.90.

2.3 Business Applications of Quadratic Functions ●

167

3. The break-even point 4. Solution of C(x) ⫽ R(x). That is, solution of 120x ⫹ 15,000 ⫽ 370x ⫺ x2, or x2 ⫺ 250x ⫹ 15,000 ⫽ 0. (x ⫺ 100)(x ⫺ 150) ⫽ 0 x ⫽ 100 or x ⫽ 150 Thus the break-even points are when 100 units and 150 units are produced.

2.3 Exercises S U P P LY, D E M A N D , A N D MARKET EQUILIBRIUM In Problems 1–4, a supply function and a demand function are given. (a) Sketch the ﬁrst-quadrant portions of those functions on the same set of axes. (b) Label the market equilibrium point. (c) Algebraically determine the market equilibrium point. 1 1. Supply: p ⫽ q2 ⫹ 10 4 Demand: p ⫽ 86 ⫺ 6q ⫺ 3q2 2. Supply: p ⫽ q2 ⫹ 8q ⫹ 16 Demand: p ⫽ 216 ⫺ 2q 3. Supply: p ⫽ 0.2q2 ⫹ 0.4q ⫹ 1.8 Demand: p ⫽ 9 ⫺ 0.2q ⫺ 0.1q2 4. Supply: p ⫽ q2 ⫹ 8q ⫹ 22 1 Demand: p ⫽ 198 ⫺ 4q ⫺ q2 4 5. If the supply function for a commodity is p ⫽ q2 ⫹ 8q ⫹ 16 and the demand function is p ⫽ ⫺3q2 ⫹ 6q ⫹ 436, ﬁnd the equilibrium quantity and equilibrium price. 6. If the supply function for a commodity is p ⫽ q2 ⫹ 8q ⫹ 20 and the demand function is p ⫽ 100 ⫺ 4q ⫺ q2, ﬁnd the equilibrium quantity and equilibrium price. 7. If the demand function for a commodity is given by the equation p2 ⫹ 4q ⫽ 1600 and the supply function is given by the equation 300 ⫺ p2 ⫹ 2q ⫽ 0, ﬁnd the equilibrium quantity and equilibrium price. 8. If the supply and demand functions for a commodity are given by 4p ⫺ q ⫽ 42 and (p ⫹ 2)q ⫽ 2100, respectively, ﬁnd the price that will result in market equilibrium. 9. If the supply and demand functions for a commodity are given by p ⫺ q ⫽ 10 and q(2p ⫺ 10) ⫽ 2100, what is the equilibrium price and what is the corresponding number of units supplied and demanded? 10. If the supply and demand functions for a certain product are given by the equations 2p ⫺ q ⫹ 6 ⫽ 0 and (p ⫹ q)(q ⫹ 10) ⫽ 3696, respectively, ﬁnd the price and quantity that give market equilibrium.

11. The supply function for a product is 2p ⫺ q ⫺ 10 ⫽ 0, while the demand function for the same product is (p ⫹ 10)(q ⫹ 30) ⫽ 7200. Find the market equilibrium point. 12. The supply and demand for a product are given by 2p ⫺ q ⫽ 50 and pq ⫽ 100 ⫹ 20q, respectively. Find the market equilibrium point. 13. For the product in Problem 11, if a $22 tax is placed on production of the item, then the supplier passes this tax on by adding $22 to his selling price. Find the new equilibrium point for this product when the tax is passed on. (The new supply function is given by p ⫽ 21q ⫹ 27.) 14. For the product in Problem 12, if a $12.50 tax is placed on production and passed through by the supplier, ﬁnd the new equilibrium point. BREAK-EVEN POINT S AND MAXIMIZATION 15. The total costs for a company are given by C(x) ⫽ 2000 ⫹ 40x ⫹ x2 and the total revenues are given by R(x) ⫽ 130x Find the break-even points. 16. If a ﬁrm has the following cost and revenue functions, ﬁnd the break-even points. 1 C(x) ⫽ 3600 ⫹ 25x ⫹ x2, 2 1 R(x) ⫽ ¢175 ⫺ x≤ x 2 17. If a company has total costs C(x) ⫽ 15,000 ⫹ 35x ⫹ 0.1x2 and total revenues given by R(x) ⫽ 385x ⫺ 0.9x2, find the break-even points. 18. If total costs are C(x) ⫽ 1600 ⫹ 1500x and total revenues are R(x) ⫽ 1600x ⫺ x2, ﬁnd the break-even points. 19. Given that P(x) ⫽ 11.5x ⫺ 0.1x2 ⫺ 150 and that production is restricted to fewer than 75 units, ﬁnd the break-even points.

168

● Chapter 2

Quadratic and Other Special Functions

20. If the proﬁt function for a ﬁrm is given by P(x) ⫽ ⫺1100 ⫹ 120x ⫺ x2 and limitations on space require that production is less than 100 units, ﬁnd the break-even points. 21. Find the maximum revenue for the revenue function R(x) ⫽ 385x ⫺ 0.9x2. 22. Find the maximum revenue for the revenue function R(x) ⫽ 1600x ⫺ x2. 23. If, in a monopoly market, the demand for a product is p ⫽ 175 ⫺ 0.50x and the revenue function is R ⫽ px, where x is the number of units sold, what price will maximize revenue? 24. If, in a monopoly market, the demand for a product is p ⫽ 1600 ⫺ x and the revenue is R ⫽ px, where x is the number of units sold, what price will maximize revenue? 25. The proﬁt function for a certain commodity is P(x) ⫽ 110x ⫺ x2 ⫺ 1000. Find the level of production that yields maximum proﬁt, and ﬁnd the maximum proﬁt. 26. The proﬁt function for a ﬁrm making widgets is P(x) ⫽ 88x ⫺ x2 ⫺ 1200. Find the number of units at which maximum proﬁt is achieved, and ﬁnd the maximum proﬁt. 27. (a) Graph the proﬁt function P(x) ⫽ 80x ⫺ 0.1x2 ⫺ 7000. (b) Find the vertex of the graph. Is it a maximum point or a minimum point? (c) Is the average rate of change of this function from x ⫽ a ⬍ 400 to x ⫽ 400 positive or negative? (d) Is the average rate of change of this function from x ⫽ 400 to x ⫽ a ⬎ 400 positive or negative? (e) Does the average rate of change of the proﬁt get closer to or farther from 0 when a is closer to 400? 28. (a) Graph the proﬁt function P(x) ⫽ 50x ⫺ 0.2x2 ⫺ 2000. (b) Find the vertex of the graph. Is it a maximum point or a minimum point? (c) Is the average rate of change of this function from x ⫽ a ⬍ 125 to x ⫽ 125 positive or negative? (d) Is the average rate of change of this function from x ⫽ 125 to x ⫽ a ⬎ 125 positive or negative? (e) Does the average rate of change of the proﬁt get closer to or farther from 0 when a is closer to 125? 29. (a) Form the proﬁt function for the cost and revenue functions in Problem 17, and ﬁnd the maximum proﬁt. (b) Compare the level of production to maximize proﬁt with the level to maximize revenue (see Problem 21). Do they agree? (c) How do the break-even points compare with the zeros of P(x)?

30. (a) Form the proﬁt function for the cost and revenue functions in Problem 18, and ﬁnd the maximum proﬁt. (b) Compare the level of production to maximize proﬁt with the level to maximize revenue (see Problem 22). Do they agree? (c) How do the break-even points compare with the zeros of P(x)? 31. Suppose a company has ﬁxed costs of $28,000 and variable costs of 25x ⫹ 222 dollars per unit, where x is the total number of units produced. Suppose further that the selling price of its product is 1250 ⫺ 35x dollars per unit. (a) Find the break-even points. (b) Find the maximum revenue. (c) Form the proﬁt function from the cost and revenue functions and ﬁnd maximum proﬁt. (d) What price will maximize the proﬁt? 32. Suppose a company has ﬁxed costs of $300 and variable costs of 34x ⫹ 1460 dollars per unit, where x is the total number of units produced. Suppose further that the selling price of its product is 1500 ⫺ 14x dollars per unit. (a) Find the break-even points. (b) Find the maximum revenue. (c) Form the proﬁt function from the cost and revenue functions and ﬁnd maximum proﬁt. (d) What price will maximize the proﬁt? 33. The following table gives the total revenues of Cablenet Communications for selected years. Year

Total Revenues (millions)

2002 2003 2004 2005 2006 2007 2008 2009 2010

$63.13 $62.91 $60.53 $60.27 $61.10 $62.19 $63.09 $64.90 $67.16

Suppose the data can be described by the equation R(t) ⫽ 0.271t2 ⫺ 2.76t ⫹ 67.83 where t is the number of years past 2000. (a) Use the function to ﬁnd the year in which revenue was minimum and ﬁnd the minimum predicted revenue. (b) Check the result from (a) against the data in the table. (c) Graph R(t) and the data points from the table. (d) Write a sentence to describe how well the function ﬁts the data.

2.4 Special Functions and Their Graphs ●

The data in the table give sales revenues and costs and expenses for Continental Divide Mining for various years. Use this table in Problems 34 and 35.

Year

Sales Revenue (millions)

Costs and Expenses (millions)

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

$2.6155 2.7474 2.934 3.3131 3.9769 4.5494 4.8949 5.1686 4.9593 5.0913 4.7489

$2.4105 2.4412 2.6378 2.9447 3.5344 3.8171 4.2587 4.8769 4.9088 4.6771 4.9025

34. Assume that sales revenues for Continental Divide Mining can be described by R(t) ⫽ ⫺0.031t2 ⫹ 0.776t ⫹ 0.179 where t is the number of years past 1997.

2.4

169

(a) Use the function to determine the year in which maximum revenue occurs and the maximum revenue it predicts. (b) Check the result from (a) against the data in the table. (c) Graph R(t) and the data points from the table. (d) Write a sentence to describe how well the function ﬁts the data. 35. Assume that costs and expenses for Continental Divide Mining can be described by C(t) ⫽ ⫺0.012t2 ⫹ 0.492t ⫹ 0.725 where t is the number of years past 1997. (a) Use R(t) as given in Problem 34 and form the proﬁt function (as a function of time). (b) Use the function from (a) to ﬁnd the year in which maximum proﬁt occurs. (c) Graph the proﬁt function from (b) and the data points from the table. (d) Through the decade from 2005 to 2015, does the function project increasing or decreasing proﬁts? Do the data support this trend (as far as it goes)? (e) How might management respond to this kind of projection?

Special Functions and Their Graphs

OBJECTIVES ●

●

●

To graph and apply basic functions, including constant and power functions To graph and apply polynomial and rational functions To graph and apply absolute value and piecewise defined functions

] Application Preview The average cost per item for a product is calculated by dividing the total cost by the number of items. Hence, if the total cost function for x units of a product is C(x) ⴝ 900 ⴙ 3x ⴙ x2 and if we denote the average cost function by C(x), then C(x) ⴝ

C(x) 900 ⴙ 3x ⴙ x2 ⴝ x x

This average cost function is a special kind of function, called a rational function. We can use this function to find the minimum average cost. (See Example 5.) In this section, we discuss polynomial, rational, and other special functions.

Basic Functions

In Section 1.3, “Linear Functions,” we saw that linear functions can be written in the form y ⫽ ax ⫹ b. The special linear function y ⫽ f(x) ⫽ x is called the identity function (Figure 2.17(a) on the next page), and a linear function deﬁned by y ⫽ f(x) ⫽ C

C a constant

is called a constant function. Figure 2.17(b) shows the graph of the constant function y ⫽ f(x) ⫽ 2. (Note that the slope of the graph of any constant function is 0.)

170

● Chapter 2

Quadratic and Other Special Functions y

y

y=x

3

y=2

3

2 1

1

x

x −3

−2

−1

1

−1

2

−3

3

−2

−1

−1

−2

−2

−3

−3

2

3

(b)

(a)

Figure 2.17

1

The functions of the form y ⫽ axb, where b ⬎ 0, are called power functions. Examples 3 of power functions include y ⫽ x2, y ⫽ x3, y ⫽ 1x ⫽ x1兾2, and y ⫽ 1 x ⫽ x1兾3. (See Fig3 ure 2.18(a)–(d).) The functions y ⫽ 1x and y ⫽ 1 x are also called root functions, and y ⫽ x2 and y ⫽ x3 are basic polynomial functions.

y

y 4 4

3

y = x3

2

3

1 2

x

y=x

1

−3

2

−2

−1

1

−1

2

3

−2 x −2

−1

1

−3

2

−1

−4

(b)

(a)

y

y 4

2

3

3

y= x

1

y= x 2

x -3

1 x -1

1

-1

1 -1

3 -2

-1

Figure 2.18

2

-2

(c)

(d)

2

3

2.4 Special Functions and Their Graphs ●

171

The general shape for the power function y ⫽ axb, where b ⬎ 0, depends on the value of b. Figure 2.19 shows the ﬁrst-quadrant portions of typical graphs of y ⫽ xb for different values of b. Note how the direction in which the graph bends differs for b ⬎ 1 and for 0 ⬍ b ⬍ 1. Getting accurate graphs of these functions requires plotting a number of points by hand or with a graphing utility. Our goal at this stage is to recognize the basic shapes of certain functions.

y

y

5

y

3

3

2

2

4 3 2

y = xb

y = x1

1

y = xb

1

1 x 1

2

3

x 1

2

3

x 1

2

b>1 (Bent upward)

b=1

0