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Visit CengageBrain.com and search for your textbook to find discounted print, digital and audio study tools that allow you to: • Study in less time to get the grade you want using online resources such as chapter quizzing, flashcards, and interactive study tools • Prepare for tests anywhere, anytime • Practice, review, and master course concepts using printed guides and manuals that work hand-in-hand with each chapter of your textbook www.CengageBrain.com Your First Study Break Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Index of Selected Applications Business and Management Accounting, 209, 520 Advertising, 9, 28, 46, 77, 94, 172–173, 267, 296, 337, 361, 365, 458, 484, 520, 555, 565, 597, 604, 611, 616, 644–645, 649, 662, 681, 725, 763, 773, 817, 826, 915, 918 Agribusiness, 237–239, 263–264, 280, 368, 889 Average cost, 46, 50, 649, 666–667, 671–672, 691, 695, 696, 865 Break-even point, 59, 63, 146–147 Budgeting, 295 Business risk, 183 Capital investment, 589, 695, 721, 731, 872, 875, 877, 890, 894, 916 Capital value, 849, 853, 866 Cellular phones, 34–35, 94, 174, 366, 475, 490, 491, 590, 611–612, 613, 617, 682 Committees, 471, 472, 487 Company growth, 28, 675–676 Competition, 235, 491, 736, 919 Construction, 279, 295, 297 Cost, 76, 77, 106–108, 119, 121, 145, 159–161, 164, 165, 247, 278, 306, 319–320, 360, 568, 579, 590, 630, 631, 634–635, 754, 777–778, 780–781, 793, 823, 826, 840, 860, 866, 873, 913, 915 Cost-benefit, 50, 178, 549–550, 555, 566, 590, 597, 691 Customer service, 818, 853 Diminishing returns, 661, 662, 695 Egg production, 207 Employee age and gender, 438–439, 447, 448 Employee benefits and compensation, 87, 256, 378, 475, 497, 905 Employee credentials, 533 Employee evaluation, 49, 476 Expense accounts, 197 Franchise growth, 764 Fraud, 439, 536 Freight logistics, 229–231 Fundraising, 124 Hospital administration, 123 Hotel pricing, 699 Human resources, 533 Industrial accidents, 441 Industrial consolidation, 337 Industrial waste, 529 Insurance, 15, 49, 476, 497, 509, 513–514, 520, 529, 532, 533 Interdepartmental costs, 247 Inventory, 197–198, 439, 529, 678–679, 682, 696, 826, 902 Job bids, 448 Job effectiveness, 49 Joint cost, 888 Labor efficiency and scheduling, 198, 307, 580 Land use, 263–264, 278, 316, 862 Maintenance costs, 458, 865 Management, 197, 198, 438–439, 440, 448, 471, 475, 533, 536, 698 Manufacturing, 77, 106, 199, 201–202, 213–214, 236, 237–238, 244–245, 251, 262–263, 266–267, 276, 277, 278, 279, 287–291, 295, 296–297, 304–305, 314–315, 316, 318, 319, 453, 520, 904, 913 Marginal cost, 619–621, 705–706, 708, 715, 741, 774–776, 879 Marginal demand, 915 Marginal profit, 621–623 Marginal return to sales, 633, 881 Marginal revenue, 107, 572, 579–580, 594–595, 618, 624, 649, 709, 715, 751, 754, 760, 763, 772 Marketing, 440 Market share, 341, 696 Material supply, 201–202 Minimizing average cost, 666–667, 671–672, 695, 696 Minimizing costs, 273–274, 301–302, 676–677, 681–682, 687, 691, 694 Oil industry, 208, 448, 457, 729–730, 754, 793, 807–808, 818, 828 Operating leverage, 183 Organizational growth, 355–356, 361 Package design, 35, 39 Parking costs, 555 Parts delivery, 448 Parts manufacturing, 244–245 Population and labor force, 187–188 Postal restrictions and rates, 77, 163, 164–165, 567, 677–678 Pricing, 83, 105, 207, 278, 323–324, 338, 509, 604, 649, 681, 699, 789, 919 Printing, 696, 919 Product design, 696 Product reliability, 336, 359, 436, 497, 508, 533, 815, 818, 850–851, 853 Production, 197–198, 208–209, 251, 273–274, 305, 314, 318–319, 611, 649, 662, 721, 725, 731, 763, 799–800, 818, 861, 875, 877–878, 889–890, 904, 910, 912–913, 915 Production costs, 165, 649, 681, 687, 698 Productivity, 487, 555, 565, 630, 649, 661, 680, 691, 695, 696, 763, 793, 795 Profit, 34, 50, 58–59, 63, 64, 76, 77, 106–107, 119–120, 121, 134, 135, 144, 256, 360, 378, 391, 554, 580, 630, 631, 691, 730–731, 780–781, 793, 826, 878 Profit maximization, 148–149, 177, 268–269, 279, 307, 319, 668–670, 672–674, 680, 695, 776–778, 899, 903, 904, 915 Property and real estate, 445, 508, 679–680, 766–767 Purchasing, 64 Purchasing electrical power, 869 Quality control, 436, 440, 457, 458, 460, 470, 473, 475, 476, 487, 495, 497, 498, 520, 529, 536, 853, 865, 866 Rentals, 34, 64, 105, 145, 221, 297, 519–520 Return to sales, 633, 881 Revenue, 34, 40, 50, 77, 106–107, 121, 145, 207, 297, 355, 360, 533, 536, 554, 555, 579, 586–587, 589, 597, 604, 608–609, 610, 616, 630, 631, 649, 712, 734–735, 777–778, 792, 793, 828, 839, 866, 913 Revenue maximization, 141–143, 177, 271, 619, 665–666, 671, 681, 682, 691, 694, 695, 737–738, 742 Rewards for employees, 475 Safety, 440, 731, 741–742 Salaries, 15, 64, 378, 422, 447, 509 Sales, 28, 46, 196, 337, 365, 438, 471, 475, 509, 520, 533, 554–555, 565, 597, 604, 611, 616, 649, 662, 680, 681, 715, 725, 773, 789, 817, 826, 915 Sales decay, 353, 359, 365 Sales growth, 360, 361 Scheduling, 277, 295, 296, 304–305, 311–312, 315, 487 Shadow prices, 278, 323–324 Shipping, 278, 296, 306–307, 322 Starbucks stores, 367, 616–617 Surplus, 865 Telecommunications, 490–491, 836, 839 Ticket sales, 221 Total income, 817, 826–827 Training, 554 Transportation, 28, 222, 236, 252, 322, 440 Unemployment rates, 507 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Unions, 146, 197, 556, 650–651 Utilities, 164, 209, 239, 245–246, 248, 487, 876, 886 Warranties, 853 Wilson’s lot size formula, 876 Wireless service spending, 64 Workers and output, 589, 610 Economics Balance of trade, 190–191 Cobb-Douglas production function, 872 Competitive market, 622, 872 Competitive products, 893 Complementary products, 893 Consumer (personal) spending, 39, 87, 94, 104, 337, 580, 715, 877 Consumer price index, 334, 338, 360, 663, 716–717, 755, 773 Consumer’s surplus, 830–831, 833–834, 835–836, 840, 865 Continuous income streams, 828, 829–830, 834–835, 840, 844, 846, 865 Demand, 39, 120, 149, 152–153, 165, 177, 354–355, 359, 526, 565, 579, 589, 602, 604, 610, 630, 709, 723, 725, 731, 732–734, 738–739, 742, 796, 826, 860–861, 915 Dependency ratio, 617 Dow Jones averages, 8–9, 66, 67, 75, 556, 674–675 Earnings and race, 87 Economic dependency ratio, 617, 650 Economic order quantity, 698 Economy models, 252 Elasticity of demand, 732–734, 738–739 Gasoline mileage and prices Gini coefficient of income, 819, 827 Gompertz curves, 355–356, 360–361, 790 Gross domestic product, 164, 173, 591, 605, 790 Gross national product, 663 Income distribution, 63, 822, 827, 846, 861, 865 Indifference curves, 874–875 Inflation, 332, 391, 741, 744, 766–767, 790–791 International trade, 196 Job factors, 9, 49 Leontief input-output models, 237–249 Lorenz curves, 819, 827 Marginal demand, 891 Marginal productivity, 889, 910 Marginal return to sales, 663 Marginal utility, 887 Market equilibrium, 109–110, 116, 120, 121, 149–150, 152–153, 177, 361, 793 Monopoly market, 669, 831–832 National consumption, 610, 778–779, 781, 793 National debt, 337, 359, 715 Oil imports, 807–808, 818, 828 Open and closed economies, 239–245, 248 Optimization in business, 280, 634 Personal income, 337–338, 488, 585–586, 605, 716, 754, 774, 809, 819, 840, 860, 866 Present value, 828, 829, 846, 860 Producer’s surplus, 832–834, 836, 840, 846, 865 Purchasing power, 332, 334, 359, 360, 364 Social security fund, 75, 127, 131–132, 135, 580, 598, 674, 675, 764, 826 Supply, 120, 177, 192–193, 359–360, 631, 709, 731, 796, 860–861 Travel and tourism, 135, 164, 682, 716, 846, 916 Unemployment, 778 Utility, 874, 877, 909, 912, 914, 915 Work force, 447 Finance Amortization, 413, 415–416, 419–420, 423, 424, 876 Annuities, 46, 394, 395, 403, 406–407, 408, 410, 423, 427–428, 565–566 Annuities due, 397, 406, 410–412 APY, 384–385 ATM transaction, 475 Banks, 84, 279, 471, 475 Bond pricing, 405–406, 412 Bonds, 510 Budgeting, 520 Capital value, 849, 853 College fund, 422, 423, 508, 509 Compound interest, 20, 336, 338, 360, 365, 383–384, 385, 386, 389–390, 741, 772, 788–789 Consumer credit and credit cards, 63, 364, 427, 477 Debt refinancing and payment, 75, 196–197 See also Loans Deferred annuities, 407–408, 412–413 Delinquent accounts, 452–453 Depreciation, 63, 79, 86, 87, 391, 817 Doubling time, 349–350, 385–386 Earnings, 94, 196, 605, 649, 905 Future value of annuities, 392–401 Future value of income stream, 829–830 Future value of investments, 329–330, 372, 379–381, 383–384, 423, 712, 714–715 Income stream, 815, 829–830 Investing, 20, 28, 35, 59, 64, 98–99, 105, 120, 218–219, 221–222, 223, 236, 251, 297–298, 327, 329–330, 340–341, 372–373, 384, 393–394, 395, 423, 427–428, 589, 604, 695, 708–709, 731, 789, 793, 876 Loans, 50, 105, 221, 413–422, 423 Mortgages and home loans, 19, 69–70, 74–75, 414–415, 427, 428, 876–877 Mutual funds, 475 Perpetuities, 565–566, 849, 853 Present value, 372, 381 Present value of annuities, 401–404, 408 Property and real estate, 445, 508, 679–680, 766–767 Purchasing power, 716 Retirement planning, 87, 475, 868, 905, 915 Savings, 779, 781, 865, 914–915 Simple interest, 39, 50, 371–373, 376–377 Sinking funds, 396, 423 Stock market, 6–7, 20–21, 338, 360, 487, 522 Taxes, 15, 16, 63, 64, 66, 67, 69, 94, 165–166, 168–169, 174, 255–256, 555, 559–560, 566, 736–738, 739–740, 742, 915–916 Trusts, 411, 412, 425, 847, 849 Venture capital, 204–205, 231–232 Life Science Adrenalin response, 566, 598 Age-sleep relationship, 87 AIDS cases, 177, 391, 447 Allometric relationships, 164, 590, 601, 731, 788, 793 Atmospheric pressure, 790 Bacterial growth, 105, 237, 327, 336, 391, 789, 793 Bee ancestry and reproduction, 237, 377 Biology, 533 Birth control, 458, 520, 540 Birth defects, 494, 496 Birth weights, 509 Blood flow, 726, 727–728, 731 Blood pressure, 529, 540, 715–716, 773 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it edition Mathematical Applications 10 for the Management, Life, and Social Sciences Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it edition Mathematical Applications 10 for the Management, Life, and Social Sciences Ronald J Harshbarger University of South Carolina, Beaufort James J Reynolds Clarion University of Pennsylvania Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it This is an electronic version of the print textbook Due to electronic rights restrictions, some third party content may be suppressed Editorial review has deemed that any suppressed content does not materially affect the overall learning experience The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Mathematical Applications for the Management, Life, and Social Sciences, Tenth Edition Ronald J Harshbarger James J Reynolds Publisher: Richard Stratton Senior Development Editor: Laura Wheel Senior Editorial Assistant: Haeree Chang Media Editor: Andrew Coppola Senior Marketing Manager: Barb Bartoszek Marketing Coordinator: Michael Ledesma Marketing Communications Manager: Mary Anne Payumo © 2013, 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 of 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 Customer & Sales Support, 1-800-354-9706 For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions Further permissions questions can be emailed to permissionrequest@cengage.com Content Project Manager: Cathy Brooks Art Director: Jill Ort Haskell Library of Congress Control Number: 2011932871 Manufacturing Planner: Doug Bertke Student Edition: ISBN-13: 978-1-133-10623-4 ISBN-10: 1-133-10623-4 Rights Acquisitions Specialist: Shalice Shah-Caldwell Production Service: Lachina Publishing Services Text Designer: Rokusek Design, Inc Cover Designer: Rokusek Design, Inc Cover Image: www.shutterstock.com Compositor: Lachina Publishing Services Instructor’s Edition: ISBN-13: 978-1-133-10847-4 ISBN-10: 1-133-10847-4 Brooks/Cole 20 Channel Center Street Boston, MA 02210 USA Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil and Japan Locate your local office at international.cengage.com/region Cengage Learning products are represented in Canada by Nelson Education, Ltd For your course and learning solutions, visit www.cengage.com Purchase any of our products at your local college store or at our preferred online store www.cengagebrain.com Instructors: Please visit login.cengage.com and log in to access instructorspecific resources Printed in the United States of America 15 14 13 12 11 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Contents Preface ix Algebraic Concepts 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Sets 3 The Real Numbers 10 Integral Exponents 16 Radicals and Rational Exponents 21 Operations with Algebraic Expressions 28 Factoring 35 Algebraic Fractions 40 2.1 2.2 2.3 2.4 2.5 Key Terms and Formulas 46 Review Exercises 48 Chapter Test 50 Extended Applications & Group Projects 52 Campaign Management 1.1 1.2 1.3 1.4 1.5 1.6 Key Terms and Formulas 116 Review Exercises 118 Chapter Test 121 Extended Applications & Group Projects 123 Hospital Administration • Fundraising Warm-up 126 Quadratic Equations 127 Quadratic Functions: Parabolas 136 Business Applications Using Quadratics 146 Special Functions and Their Graphs 153 Modeling; Fitting Curves to Data with Graphing Utilities (optional) 165 Key Terms and Formulas 175 Review Exercises 176 Chapter Test 179 Extended Applications & Group Projects 181 An Inconvenient Truth • Body-Mass Index (Modeling) • Operating Leverage and Business Risk Linear Equations and Functions 53 Warm-up 54 Solutions of Linear Equations and Inequalities in One Variable 55 Functions 65 Linear Functions 78 Graphs and Graphing Utilities 88 Solutions of Systems of Linear Equations 95 Applications of Functions in Business and Economics 106 Quadratic and Other Special Functions 125 3.1 3.2 3.3 3.4 3.5 Matrices 184 Warm-up 185 Matrices 186 Multiplication of Matrices 199 Gauss-Jordan Elimination: Solving Systems of Equations 209 Inverse of a Square Matrix; Matrix Equations 224 Applications of Matrices: Leontief InputOutput Models 237 Key Terms and Formulas 249 Review Exercises 250 Chapter Test 252 Extended Applications & Group Projects 255 Taxation • Company Profits after Bonuses and Taxes v Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.freebookslides.com Section 1.2 Functions 67 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) 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 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 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 For example, if x 8, then y or y 24, 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) 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 Quadrant I A(5, 2) B(−4, 1) −5 −4 −3 −2 −1 −1 x-axis −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 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 68 Chapter Linear Equations and Functions www.freebookslides.com Example 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 2 y 1 16 14 12 10 y = 2x x −1 y = 4x 2 −4 −3 −2 −1 −2 4 −3 −4 (a) Figure 1.5 −2 x 1 (b) 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 Function Notation 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 intersects the curve at (2, 2) and 2, 22 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 then and f x 3x 2x 1 f 2 2 2 2 1 17 f 23 23 2 23 1 22 Figure 1.6 shows the function notation f (x) as (a) an operator on x and (b) a y-coordinate for a given x-value Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.freebookslides.com Section 1.2 Functions 69 y (−3, f (−3)) input x y = f(x) f −3 −2 −1 output f(x) −1 x (2, f(2)) (b) (a) Figure 1.6 (0, f (0)) Letters other than f may also be used to denote functions For example, y g x or y h x may be used Example Evaluating Functions If y f x 2x3 3x2 1, find the following (a) f (b) f 21 (c) f 2a Solution (a) f 3 3 2 1 27 2 1 28 Thus y 28 when x (b) f 21 21 3 21 2 1 21 2 1 1 24 Thus y 24 when x 21 (c) f 2a 2a 3 2a 2 1 22a3 3a2 1 ■ When a function fits a set of data exactly or approximately, we say that the function models the data The model includes descriptions of all involved variables Example Electronic Income Tax Returns | Application Preview | The relationship between the number y of individual income tax returns filed electronically and the number of years after 1995 can be modeled by the function 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 part (a) Solution (a) f 5.091 11.545 52.273 (b) The statement f 52.273 means that in 1995 2003, 52.273 million income tax returns were filed electronically. ■ Example Mortgage Payment | Table 1.2 | r(%) f(r) 2.6 5.2 6.3 7.4 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 # f 2.6 f # 2.6 ? Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 70 Chapter Linear Equations and Functions www.freebookslides.com 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 30, so r (c) 2 # f 2.6 # 12 24 and f # 2.6 f 5.2 15, so # f 2.6 2 f # 2.6 ■ Example Function Notation Given f x x2 3x 8, find f 1x h2 f 1x2 h and simplify (if h 0) Solution We find f x h by replacing each x in f (x) with the expression x h f 1x h2 f 1x2 h 5 Domains and Ranges x h 2 x h x2 3x h x2 2xh h2 2 3x 3h x2 3x h x2 2xh h2 3x 3h x2 3x h h 2x h 2xh h2 3h 5 2x h 3 h h ■ 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 values that result in a denominator of 0, and values that result in an even root of a negative number Example Domain and Range Find the domain of each of the following functions; find the range for the functions in parts (a) and (b) (a) y 4x2 (b) y "4 x (c) y 1 x22 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 $ The graph shown in Figure 1.7(a) illustrates our conclusions about the domain and range (b) We note the restriction that x cannot be negative Thus the domain consists of only numbers less than or equal to That is, the domain is the set of real numbers satisfying x # Because !4 x is always nonnegative, the range is all y $ Figure 1.7(b) shows the graph of y !4 x Note that the graph is located only where x # and on or above the x-axis (where y $ 0) 1 (c) y 1 is undefined at x because is undefined Hence, the domain x22 consists of all real numbers except Figure 1.7(c) shows the graph of y 1 x22 The break where x indicates that x is not part of the domain. ■ Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.freebookslides.com Section 1.2 y Functions 71 y 14 y=1+ 12 10 x−2 x −8 −6 −4 −2 −2 y = 4x 2 −2 −4 −6 x −4 −3 −2 −1 4 −8 (b) (a) (c) Figure 1.7 Checkpoint 1. If y f x , the independent variable is _ and the dependent variable is _ 2. If (1, 3) is on the graph of y f x , then f 1 ? 3. If f x x3, find f 22 4. If f x 2x2, find f x h , what is the domain of f (x)? 5. If f x x11 Operations with Functions 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 Operations with Functions Let f and g be functions of x, and define the following Sum Difference Product Quotient f g2 1x2 f g2 1x2 f # g2 1x2 f a b 1x2 g f 1x2 f 1x2 f 1x2 f 1x2 g1x2 g1x2 g1x2 # g1x2 if g x 2 Example Operations with Functions If f x 3x and g x x2 3, find the following functions f (a) f g x (b) f g x (c) f # g x (d) a b x g Solution (a) f g x f x g x 3x 2 1 x2 x2 3x (b) f g x f x 2 g x 3x 2 x2 2x2 3x (c) f # g x f x # g x 3x 2 x2 3x3 2x2 9x f 1x2 f 3x (d) a b x 5 , if x2 0 g x 23 g1x2 ■ 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 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 72 Chapter www.freebookslides.com Linear Equations and Functions Composite Functions Let f and g be functions Then the composite functions g of f (denoted g + f ) and f of g (denoted f + g) are defined as follows: 1g + f 1x2 g1 f 1x2 f + g2 1x2 f 1g1x2 Note that the domain of g + f is the subset of the domain of f for which g + f is defined Similarly, the domain of f + g is the subset of the domain of g for which f + g is defined Example Composite Functions If f x 2x3 1 and g x x2, find the following (a) g + f x (b) f + g x Solution (a) g + f x g f x 2 g 2x3 1 2x3 1 2 4x6 4x3 1 (b) f + g x f g x 2 f x2 x2 1 2x6 1 ■ Figure 1.8 illustrates both composite functions found in Example (g f )(x) x g f(x) = 2x + g Checkpoint Checkpoint solutions g)(x) x f Figure 1.8 (f g( f(x)) = (2x + 1)2 (g f )(x) = 4x + 4x + g(x) = x f f(g(x)) = 2(x 2)3 + ( f g)(x) = 2x + 6. If f x 2x and g x 3x2, find the following (a) g f x (b) f # g x (c) f + g x (d) g + f x (e) f + f x f f x 2 1. Independent variable is x; dependent variable is y 2. f 1 3. f 22 22 28 4. f x h x h 2 x2 2xh h2 5. The domain is all real numbers except x 21, because f x is undefined when x 21 6. (a) g f x g x 2 f x 3x2 1 2x 3x2 2x (b) f # g x f x # g x 1 2x 3x2 3x2 6x3 (c) f + g x f g x 2 f 3x2 2 3x2 6x2 (d) g + f x g f x 2 g 1 2x 1 2x 2 (e) f + f x f f x 2 f 1 2x 2 1 2x 4x Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.freebookslides.com Section 1.2 Functions 73 | Exercises | 1.2 In Problems and 2, use the values in the following table x –7 –1 4.2 11 14 18 22 y 0 11 35 22 22 60 (a) Explain why the table defines y as a function of x (b) State the domain and range of this function (c) If the table expresses y f x , find f (0) and f (11) (a) 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 and 4, are the relations defined by the tables functions? Explain why or why not and give the domain and range x y –4 –4 16 x –1 y Do either of the graphs in Figure 1.9 represent y as a function of x? Explain your choices y y (9, 10) 12 10 4 (5, 6) 2 10 12 -4 -6 x -2 -2 (2, 2) (a) 20 x -6 (b) Figure 1.9 Do either of the graphs in Figure 1.10 represent y as a function of x? Explain your choices y x -4 -2 -4 (a) Figure 1.10 21 y 10 11 12 13 14 15 16 17 18 19 22 10 -8 -4 -4 -6 -8 -10 x 10 23 24 (b) 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 23 (d) R(1.6) If f x 17 6x, find the following (a) f 23 (b) f (1) (c) f (10) (d) f 23 2 If C x 4x 3, find the following (a) C(0) (b) C 21 (c) C 22 (d) C 1232 2 If h x 3x 2x, find the following (a) h(3) (b) h 23 (c) h(2) (d) h 16 If h x x 2 x 3, find the following (c) h(6) (d) h(2.5) (a) h 21 (b) h(0) If R x 100x x3, find the following (a) R(1) (b) R(10) (c) R(2) (d) R 210 If f x x3 4/x, find the following (a) f 1212 (b) f (2) (c) f 22 If C x x2 2 /x, find the following (a) C(1) (b) C 12 (c) C 22 Let f x 1 x x2 and h (a) Is f 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 1x h2 f 1x2 (e) Find and simplify h Let f x 3x2 6x and h (a) Is f 2 f 2? (b) Find f x h (c) Does f x h f x h? (d) Does f x h f x f h ? f 1x h2 f 1x2 (e) Find and simplify h If f x x 2x2 and h 0, find the following and simplify f 1x h2 f 1x2 (a) f x h (b) h If f x 2x2 x and h 0, find the following and simplify f 1x h2 f 1x2 (a) f x h (b) h If y f x in Figure 1.9(a), find the following (a) f (9) (b) f (5) Suppose y g x in Figure 1.10(b) (a) Find g(0) (b) How many x-values in the domain of this function satisfy g x 0? Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 74 Chapter Linear Equations and Functions www.freebookslides.com 25 The graph of y x2 4x is shown in Figure 1.11 y y = x – 4x P(a, b) In Problems 31–34, a function and its graph are given In each problem, find the domain "x x11 32 f x 31 f x x22 "x y y x -1 10 10 5 -2 Q x x R -4 10 12 Figure 1.11 (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? 26 The graph of y 2x2 is shown in Figure 1.12 -10 10 34 f x 22 "9 x2 y y x 12 -4 -4 -2 -2 -4 x -8 33 f x "49 x2 –5 -5 -4 2 -6 y P(a, b) y = 2x R x -3 -2 -1 Figure 1.12 (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 2x2? State the domain and range of each of the functions in Problems 27–30 27 y x2 28 y x2 29 y "x 30 y "x2 1 For f (x) and g(x) given in Problems 35–38, find (a) f g x (b) f g x (c) f # g x (d) f yg x 35 f x 3x g x x3 36 f x "x g x 1/x 37 f x "2x g x x2 38 f x x 2 g x 2x For f (x) and g(x) given in Problems 39–42, find (a) f + g x (b) g + f x (c) f f x 2 (d) f x f # f x 39 f x x 2 g x 2x 40 f x 3x g x x3 41 f x 2"x g x x4 42 f x g x 4x 1 x APPLICATIONS 43 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 to repay the Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.freebookslides.com debt If we denote this function by A f t , then the following table defines the function A t A 5 40,000 20 103,000 10 69,000 25 113,000 15 89,000 30 120,000 18 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 f f ? 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 1t2 44 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 a couple’s 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 1980.12 15 898.83 10 1161.09 20 775.30 12 1028.39 25 706.78 Source: Comprehensive Mortgage Payment Tables, Publication No 492, Financial Publishing Co., Boston (a) If they refinance for 20 years, what is the monthly payment? Write this correspondence in the form R f 1t2 (b) Find f (10) and write a sentence that explains its meaning (c) Is f 5 f f ? Explain 45 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 1t2 (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 were graphed, what parts of the new graph must be the same as this graph and what parts might be the same? Explain Functions 75 (d) Find the domain and range of n f t if the function is defined by the graph Workers per beneficiary t Section 1.2 16 16.5 14 12 10 3.4 1.9 1950 1970 1990 2010 2030 2050 Year Source: Social Security Administration 46 Dow Jones Industrial Average The graph shows the Dow Jones Industrial Average on a particularly tumultuous day soon after the U.S credit rating was downgraded from AAA to AA1 in August of 2011 If t represents the number of hours after 9:30 a.m., 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 11,000? Estimate one such t-value Dow Jones Industrial Average 11,300 11,200 11,100 11,000 10,900 10,800 Down 508.06 Close = 10,719.94 10,700 10,600 10 a.m Noon p.m p.m Source: Google Finance, August 10, 2011 47 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 number of prisoners and y g t gives the number of parolees, use the figure to complete the following (a) Estimate f (105) and g(105) (b) Find f (107) and explain its meaning (c) Find g(92) and explain its meaning (d) Find f g 107 and explain its meaning (e) Which of f g 95 and f g 105 is greater? Explain Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 76 Chapter Linear Equations and Functions 50 Temperature measurement The equation Imprisonment outpaces release on parole Number in state prisons and on parole annually at year's end: 1.6 www.freebookslides.com in prison on parole 160 C5 F2 9 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°? 51 Cost The total cost of producing a product is given by (in millions) 0.85 1.2 1.51 1.33 0.8 0.4 0.73 0.66 0.82 0.0 ’90 ’92 ’94 ’96 ’98 ’00 ’01 ’02 ’03 ’04 ’05 ’06 ’07 Source: U.S Department of Justice Number of women in the work force (millions) 48 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 and projected to 2016, is shown in the figure (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 2005 2 f 1990 and explain its meaning 80 70 60 50 40 30 20 10 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) 52 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) 53 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 C1p2 1920 1940 1960 1980 Year 2000 2020 Source: U.S Bureau of Labor Statistics 49 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 original work, if the air temperature is 25°F, then the wind chill, WC, is a function of the wind speed, s (in mph), and is given by WC f s 45.694 1.75s 29.26!s (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 $ How can you tell that s is not in the working domain, even though it is in the mathematical domain? 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) 54 Test reliability If a test that has reliability r is lengthened by a factor n n $ , the reliability R of the new test is given by R1n2 nr 1 1n 12r 0,r#1 If the reliability is r 0.6, the equation becomes R1n2 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? Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.freebookslides.com 55 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? 56 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? 57 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 + 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 58 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 # t # 20 Section 1.2 Functions 77 Find W + L t in order to express W as a function of the age t of the fish 59 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 + g defined? Explain (b) Is g + f defined? Explain (c) For the functions in parts (a) and (b) that are defined, identify the input (independent variable) and the output (dependent variable) and explain what the function means 60 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 + s door (d) s + p door (e) p + p door 61 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 62 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 63 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 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 64 Revenue A company handles an apartment building with 50 units Experience has shown that if the rent for 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 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 78 Chapter Linear Equations and Functions Objectives www.freebookslides.com 1.3 • To find the intercepts 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 Linear Functions | Application Preview | The number of banks in the United States for selected years from 1980 to 2009 is given by y 2380.961x 18,483.167 where x is the number of years after 1980 (Source: Federal Deposit Insurance Corporation) What does this function tell about how the number of banks has changed per year during this period? (See Example 7.) 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, defined as follows A linear function is a function of the form where a and b are constants y f x ax b 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 and any point on the y-axis has x-coordinate 0, we find intercepts as follows Intercepts (a) To find the y-intercept(s) of the graph of an equation, set x 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 and solve for x Example Intercepts Find the intercepts and graph the following (a) 3x y 9 (b) x 4y Solution (a) To find the y-intercept, we set x and solve for y: y gives y 9, so the y-intercept is To find the x-intercept, we set y and solve for x: 3x gives x 3, so the x-intercept is Using the intercepts gives the graph, shown in Figure 1.13 (b) Letting x gives y 0, and letting y 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 in x 4y, we get x 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 1.14 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.freebookslides.com Section 1.3 y y 10 3x + y = (0, 9) x = 4y −10 −8 −6 −4 −2 (4,1) (3, 0) x −4 10 −1 −1 −5 −2 x (0, 0) −3 −10 Figures 1.13 & 1.14 Linear Functions 79 −4 Note that the equation graphed in Figure 1.13 can be rewritten as y 3x or f x 3x We see in Figure 1.13 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 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 gives 4096x 491,520 x 120 Thus 120 is the x-intercept y-intercept: x gives 4y 491,520 y 122,880 Thus 122,880 is the y-intercept The graph is shown in Figure 1.15 Note that the units on the x- and y-axes are different and that the y-intercept corresponds to the value of the property 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 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 20 Figure 1.15 40 60 80 100 120 140 x Months Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 80 Chapter www.freebookslides.com Linear Equations and Functions Section 1.3 Linear Functions 80 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; Note that in Figure 1.15, as the graph moves from the y-intercept point (0, 122,880) to the Slope of a Line x-intercept point (120, 0), the y-value on the line changes 2122,880 units (from 122,880 to 0), whereas the x-value changes 120 units (from to 120) Thus the rate of change of the value of the business property is 2122,880 21024 dollars per month 120 This means that each month the value of the property changes by 21024 dollars, or the value decreases by $1024 per month This rate of change of a linear function is called the slope of the line that is its graph (see Figure 1.15) 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 m5 y2 y1 Dy x2 x1 Dx where Dy, read “delta y,” means “change in y” and Dx means “change in x.” The slope of a vertical line is undefined y P2(x2, y2) y2 y1 y2 − y1 P1(x1, y1) x2 − x1 x1 x2 x 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 Example Slopes Find the slope of (a) line ,1, passing through 22, and (4, 3) (b) line ,2, passing through (3, 0) and 4, 23 Solution 321 123 22 5 or, equivalently, m 5 22 26 22 This means that a point units to the right and unit up from any point on the line is also on the line Line ,1 is shown in Figure 1.16 (a) m y (4, 3) (−2, 1) (3, 0) −3 −2 −1 x −2 (4, −3) Figure 1.16 −4 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.freebookslides.com Section 1.3 Linear Functions 81 23 5 23 324 21 This means that a point unit to the right and units down from any point on the line is also on the line Line ,2 is also shown in Figure 1.16 (b) m 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 y m5 Dy Dx m>0 3. The slope of a horizontal line is 0, because Dy The function is constant y m5 x 2. The slope is negative if the line slopes downward toward the right The function is decreasing Dy Dx 50 Dy Dx x 4. The slope of a vertical line is undefined, because Dx y m5 m=0 y ,0 x m5 Dy Dx is undefined x m