To the Student As you begin, you may feel anxious about the number of theorems, definitions, procedures, and equations You may wonder if you can learn it all in time Don’t worry—your concerns are normal This textbook was written with you in mind If you attend class, work hard, and read and study this text, you will build the knowledge and skills you need to be successful Here’s how you can use the text to your benefit Read Carefully When you get busy, it’s easy to skip reading and go right to the problems Don’t the text has a large number of examples and clear explanations to help you break down the mathematics into easy-to-understand steps Reading will provide you with a clearer understanding, beyond simple memorization Read before class (not after) so you can ask questions about anything you didn’t understand You’ll be amazed at how much more you’ll get out of class if you this Use the Features I use many different methods in the classroom to communicate Those methods, when incorporated into the text, are called “features.” The features serve many purposes, from providing timely review of material you learned before (just when you need it) to providing organized review sessions to help you prepare for quizzes and tests Take advantage of the features and you will master the material To make this easier, we’ve provided a brief guide to getting the most from this text Refer to “Prepare for Class,” “Practice,” and “Review” on the following three pages Spend fifteen minutes reviewing the guide and familiarizing yourself with the features by flipping to the page numbers provided Then, as you read, use them This is the best way to make the most of your text Please not hesitate to contact me, through Pearson Education, with any questions, comments, or suggestions for improving this text I look forward to hearing from you, and good luck with all of your studies Best Wishes! Prepare for Class ‘‘Read the Book’’ Feature Description Benefit Page Every Chapter Opener begins with Chapter- Opening Each chapter begins with a discussion of a topic of current interest and ends with a Topic & Project The Project lets you apply what you learned to solve a problem related to the topic 402 The projects allow for the integration of spreadsheet technology that you will need to be a productive member of the workforce The projects give you an opportunity to collaborate and use mathematics to deal with issues of current interest 503 Each section begins with a list of objectives Objectives also appear in the text where the objective is covered These focus your studying by emphasizing what’s most important and where to find it 423 Preparing for this Section Most sections begin with a list of key concepts to review with page numbers Ever forget what you’ve learned? This feature highlights previously learned material to be used in this section Review it, and you’ll always be prepared to move forward 423 Now Work the ‘Are You Prepared?’ Problems Problems that assess whether you have the Not sure you need the Preparing for This 423, 434 prerequisite knowledge for the upcoming Section review? Work the ‘Are You section Prepared?’ problems If you get one wrong, you’ll know exactly what you need to review and where to review it! Now Work These follow most examples and direct you to a related exercise related project Internet-Based Projects Every Section begins with Learning Objectives Sections contain problems WARNING Exploration and Seeing the Concept In Words Calculus SHOWCASE EXAMPLES Model It! Examples and Problems Warnings are provided in the text We learn best by doing You’ll solidify your understanding of examples if you try a similar problem right away, to be sure you understand what you’ve just read These point out common mistakes and help you to avoid them 430, 435 456 418, 443 These graphing utility activities foreshadow a concept or solidify a concept just presented You will obtain a deeper and more intuitive understanding of theorems and definitions These provide alternative descriptions of select definitions and theorems Does math ever look foreign to you? This feature translates math into plain English These appear next to information essential for the study of calculus Pay attention–if you spend extra time now, you’ll better later! These examples provide “how-to” instruction by offering a guided, step-by-step approach to solving a problem With each step presented on the left and the mathematics displayed on the right, you can immediately see how each step is employed 334 These examples and problems require you to build a mathematical model from either a verbal description or data The homework Model It! problems are marked by purple headings It is rare for a problem to come in the form “Solve the following equation.” Rather, the equation must be developed based on an explanation of the problem These problems require you to develop models that will allow you to describe the problem mathematically and suggest a solution to the problem 447, 475 440 205, 407, 431 Feature Practice ‘‘Work the Problems’’ Description Benefit Page ‘Are You Prepared?’ Problems These assess your retention of the prerequisite material you’ll need Answers are given at the end of the section exercises This feature is related to the Preparing for This Section feature Do you always remember what you’ve learned? Working these problems is the best way to find out If you get one wrong, you’ll know exactly what you need to review and where to review it! 434, 440 Concepts and Vocabulary These short-answer questions, mainly Fill-in-the-Blank, Multiple-Choice and True/False items, assess your understanding of key definitions and concepts in the current section It is difficult to learn math without knowing the language of mathematics These problems test your understanding of the formulas and vocabulary 434 Skill Building Correlated with section examples, these problems provide straightforward practice It’s important to dig in and develop your skills These problems provide you with ample opportunity to so 434–436 Mixed Practice These problems offer comprehensive assessment of the skills learned in the section by asking problems that relate to more than one concept or objective These problems may also require you to utilize skills learned in previous sections Learning mathematics is a building process Many concepts are interrelated These problems help you see how mathematics builds on itself and also see how the concepts tie together 436–437 Applications and Extensions These problems allow you to apply your skills to real-world problems They also allow you to extend concepts learned in the section You will see that the material learned within the section has many uses in everyday life 437–439 Explaining Concepts: “Discussion and Writing” problems are colored red They support class Discussion and discussion, verbalization of mathematical Writing To verbalize an idea, or to describe it clearly in writing, shows real understanding These problems nurture that understanding Many are challenging, but you’ll get out what you put in 439 NEW! Retain Your Knowledge These problems allow you to practice content learned earlier in the course Remembering how to solve all the different kinds of problems that you encounter throughout the course is difficult This practice helps you remember 439 Now Work Many examples refer you to a related homework problem These related problems are marked by a pencil and orange numbers If you get stuck while working problems, look for the closest Now Work problem, and refer to the related example to see if it helps Every chapter concludes with a comprehensive list of exercises to pratice Use the list of objectives to determine the objective and examples that correspond to the problems Work these problems to ensure that you 499–501 understand all the skills and concepts of the chapter Think of it as a comprehensive review of the chapter ideas, and writing and research projects problems Review Exercises 432, 435, 436 Review ‘‘Study for Quizzes and Tests’’ Feature Description Benefit Page The Chapter Review at the end of each chapter contains Things to Know A detailed list of important theorems, formulas, and definitions from the chapter Review these and you’ll know the most important material in the chapter! You Should Be Able to Contains a complete list of objectives by section, examples that illustrate the objective, and practice exercises that test your understanding of the objective Do the recommended exercises and you’ll 498–499 have mastered the key material If you get something wrong, go back and work through the example listed and try again Review Exercises These provide comprehensive review and Practice makes perfect These problems 499–501 practice of key skills, matched to the Learning combine exercises from all sections, giving you a comprehensive review in one Objectives for each section place Chapter Test About 15–20 problems that can be taken Be prepared Take the sample practice as a Chapter Test Be sure to take the Chapter test under test conditions This will get you ready for your instructor’s test If you get a Test under test conditions—no notes! problem wrong, you can watch the Chapter Test Prep Video Cumulative Review These problem sets appear at the end of each chapter, beginning with Chapter They combine problems from previous chapters, providing an ongoing cumulative review When you use them in conjunction with the Retain Your Knowledge problems, you will be ready for the final exam These problem sets are really important 502–503 Completing them will ensure that you are not forgetting anything as you go This will go a long way toward keeping you primed for the final exam Chapter Projects The Chapter Projects apply to what you’ve learned in the chapter Additional projects are available on the Instructor’s Resource Center (IRC) The Chapter Projects give you an opportunity 503–504 to apply what you’ve learned in the chapter to the opening topic If your instructor allows, these make excellent opportunities to work in a group, which is often the best way of learning math Internet-Based In selected chapters, a Web-based project These projects give you an opportunity to is given collaborate and use mathematics to deal with issues of current interest by using the Internet to research and collect data Projects 497–498 502 503 Achieve Your Potential The author, Michael Sullivan, has developed specific content in MyMathLab® to ensure you have many resources to help you achieve success in mathematics - and beyond! The MyMathLab features described here will help you: • Review math skills and concepts you may have forgotten • Retain new concepts as you move through your math course • Develop skills that will help with your transition to college Adaptive Study Plan The Study Plan will help you study more efficiently and effectively Your performance and activity are assessed continually in real time, providing a personalized experience based on your individual needs Skills for Success The Skills for Success Modules support your continued success in college These modules provide tutorials and guidance on a variety of topics, including transitioning to college, online learning, time management, and more Additional content is provided to help with the development of professional skills such as resume writing and interview preparation Getting Ready Are you frustrated when you know you learned a math concept in the past, but you can’t quite remember the skill when it’s time to use it? Don’t worry! The author has included Getting Ready material so you can brush up on forgotten material efficiently by taking a quick skill review quiz to pinpoint the areas where you need help Then, a personalized homework assignment provides additional practice on those forgotten concepts, right when you need it Retain Your Knowledge As you work through your math course, these MyMathLab® exercises support ongoing review to help you maintain essential skills The ability to recall important math concepts as you continually acquire new mathematical skills will help you be successful in this math course and in your future math courses College Algebra Tenth Edition Michael Sullivan Chicago State University Boston Columbus Indianapolis New York San Francisco Hoboken Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo Editor in Chief: Anne Kelly Acquisitions Editor: Dawn Murrin Assistant Editor: Joseph Colella Program Team Lead: Marianne Stepanian Program Manager: Chere Bemelmans Project Team Lead: Peter Silvia Project Manager: Peggy McMahon Associate Media Producer: Marielle Guiney Senior Project Manager, MyMathLab: Kristina Evans QA Manager, Assessment Content: Marty Wright Senior Marketing Manager: Michelle Cook Marketing Manager: Peggy Sue Lucas Marketing Assistant: Justine Goulart Senior Author Support/Technology Specialist: Joe Vetere Procurement Manager: Vincent Scelta Procurement Specialist: Carol Melville Text Design: Tamara Newnam Production Coordination, Composition, Illustrations: Cenveo® Publisher Services Associate Director of Design, USHE EMSS/HSC/EDU: Andrea Nix Project Manager, Rights and Permissions: Diahanne Lucas Dowridge Art Director: Heather Scott Cover Design and Cover Illustration: Tamara Newnam Acknowledgments of third-party content appear on page C1, which constitutes an extension of this copyright page Unless otherwise indicated herein, any third-party trademarks that may appear in this work are the property of their respective owners, and any references to third-party trademarks, logos or other trade dress are for demonstrative or descriptive purposes only Such references are not intended to imply any sponsorship, endorsement, authorization, or promotion of Pearson’s products by the owners of such marks, or any relationship between the owner and Pearson Education, Inc or its affiliates, authors, licensees or distributors Microsoft® and Windows® are registered trademarks of the Microsoft Corporation in the U.S.A and other countries Screen shots and icons reprinted with permission from the Microsoft Corporation This book is not sponsored or endorsed by or affiliated with the Microsoft Corporation Microsoft and /or its respective suppliers make no representations about the suitability of the information contained in the documents and related graphics published as part of the services for any purpose All such documents and related 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described herein at any time Partial screen shots may be viewed in full within the software version specified The student edition of this text has been cataloged as follows: Library of Congress Cataloging-in-Publication Data Sullivan, Michael, 1942College algebra / Michael Sullivan, Chicago State University Tenth edition pages cm ISBN 978-0-321-97947-6 Algebra Textbooks Algebra-Study and teaching (Higher) I Title QA154.3.S763 2016 512.9 dc23 2014021757 Copyright © 2016 by Pearson Education, Inc or its affiliates All Rights Reserved Printed in the United States of America This publication is protected by copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise For information regarding permissions, request forms, and the appropriate contacts within the Pearson Education Global Rights & Permissions department, please visit www.pearsoned.com/permissions/ PEARSON, ALWAYS LEARNING, and MYMATHLAB are exclusive trademarks in the U.S and/or other countries owned by Pearson Education, Inc or its affiliates 10—CRK—17 16 15 14 www.pearsonhighered.com ISBN-10: 0-321-97947-8 ISBN-13: 978-0-321-97947-6 Contents Three Distinct Series xvi The Contemporary Series xvii Preface to the Instructor xviii Resources for Success xxii Applications Index xxv R Review 1 R.1 Real Numbers 2 Work with Sets • Classify Numbers • Evaluate Numerical Expressions • Work with Properties of Real Numbers R.2 Algebra Essentials 17 Graph Inequalities •Find Distance on the Real Number Line • Evaluate Algebraic Expressions • Determine the Domain of a Variable • Use the Laws of Exponents • Evaluate Square Roots • Use a Calculator to Evaluate Exponents • Use Scientific Notation R.3 Geometry Essentials 30 Use the Pythagorean Theorem and Its Converse • Know Geometry Formulas • Understand Congruent Triangles and Similar Triangles R.4 Polynomials 39 Recognize Monomials • Recognize Polynomials • Add and Subtract Polynomials • Multiply Polynomials • Know Formulas for Special Products • Divide Polynomials Using Long Division • Work with Polynomials in Two Variables R.5 Factoring Polynomials 49 Factor the Difference of Two Squares and the Sum and Difference of Two Cubes • Factor Perfect Squares • Factor a Second-Degree Polynomial: x2 + Bx + C • Factor by Grouping • Factor a Second-Degree Polynomial: Ax2 + Bx + C, A ≠ • Complete the Square R.6 Synthetic Division 58 Divide Polynomials Using Synthetic Division R.7 Rational Expressions 62 Reduce a Rational Expression to Lowest Terms • Multiply and Divide Rational Expressions • Add and Subtract Rational Expressions • Use the Least Common Multiple Method • Simplify Complex Rational Expressions R.8 nth Roots; Rational Exponents 73 Work with nth Roots • Simplify Radicals • Rationalize Denominators • Simplify Expressions with Rational Exponents Equations and Inequalities 81 1.1 Linear Equations 82 Solve a Linear Equation • Solve Equations That Lead to Linear Equations • Solve Problems That Can Be Modeled by Linear Equations 1.2 Quadratic Equations 92 Solve a Quadratic Equation by Factoring • Solve a Quadratic Equation by Completing the Square • Solve a Quadratic Equation Using the Quadratic Formula • Solve Problems That Can Be Modeled by Quadratic Equations vii Answers Section 5.3 AN29 1x + 42 1x - 32 33 R1x2 = 1x + 22 1x - 32 ; Domain: 5x͉ x ≠ - 2, x ≠ 2 In lowest terms, R1x2 = x + 3 y-intercept: 2; x-intercept: - x + 4 Vertical asymptote: x = - 2; hole at a3, b 5 Horizontal asymptote: y = 1, not intersected ؊4 ؊2 6 7 Interval (؊ؕ, ؊4) Number Chosen ؊5 Value of R R(؊5) ؍ 1– Location of Graph Above x-axis Point on Graph 35 R1x2 = (؊5, ) 1– 13x + 12 12x - 32 1x - 22 12x - 32 ؊5, (؊4, ؊2) (؊2, 3) (3, ؕ) ؊3 R(؊3) ؍؊1 R(0) ؍2 R(4) – ؍3 Below x-axis Above x-axis Above x-axis ; Domain: e x ` x ≠ (؊4, 0) Interval (؊ؕ, ) ( Number Chosen ؊1 Value of R R(؊1) ؍23 Location of Graph Above x-axis Point on Graph (؊1, ) 3 ؊3 , ) (4, ) 3x + 1 , x ≠ f 2 In lowest terms, R1x2 = 3 y-intercept: - ; x-intercept: x - 2 R(0) ؍ ( 2) (2, ؕ) 1.7 R(1.7) Ϸ ؊20.3 R(6) ؍4.75 Below x-axis Above x-axis (1.7, ؊20.3) (6, 4.75) 2, ؊2 Below x-axis (0, ؊ ) 4, x ؍؊2 4 Vertical asymptote: x = 2; hole at a , - 11 b 5 Horizontal asymptote: y = 3, not intersected ؊3 2 6 7 ؊3 3, (0, 2) y؍1 10 x (؊3, ؊1) 4– (0, 2) (؊3, ؊1) y ؊1, y x؍2 (6, 4.75) y؍3 ؊ 10 x , ؊11 ,0 0, ؊ 1x + 32 1x + 22 ; Domain: 5x͉ x ≠ - 2 In lowest terms, R1x2 = x + 2 3 y-intercept: 2; x-intercept: - x + 4 Vertical asymptote: none; hole at - 3, - 12 5 Oblique asymptote: y = x + intersected at all points except x = - y ؊3 ؊2 6 7 37 R1x2 = Interval (؊ؕ, ؊3) (؊3, ؊2) Number Chosen ؊4 ؊2 Value of R R(؊4) ؍؊2 Location of Graph Below x-axis Point on Graph (؊4, ؊2) R (؊2, ؕ) ( )؍ ؊2 R(0) ؍2 Below x-axis Above x-axis (؊ (0, 2) ) 5, ؊2 (0, 2) (؊2, 0) (؊3, ؊1) ؊2 x ؊ ,؊ 2 (؊4, ؊2) - 1x - 22 -3 ; Domain: 5x͉ x ≠ - 2, x ≠ 2 In lowest terms, H1x2 = 3 y-intercept: - ; no x-intercept 1x - 22 1x + 22 x + 2 4 Vertical asymptote: x = - 2; hole at a2, - b 5 Horizontal asymptote: y = 0; not intersected ؊2 x = −2 y 6 7 Q2, − 3R 39 H1x2 = Interval Number Chosen Value of H (؊ؕ, ؊2) ؊3 H(؊3) ؍3 Location of Graph Above x-axis Point on Graph (؊3, 3) (؊2, 2) 3 H(0) ؍؊ H(3) ؍؊ Below x-axis Below x-axis (0 , ؊ ) (3 , ؊ ) 10 (2, ؕ) Q3, − 35R (−3, 3) y=0 −7 x Q0, − 32R 1x - 12 1x - 42 x - ; Domain: {x͉ x ≠ 1} 2 In lowest terms, F1x2 = 3 y-intercept: 4; x-intercept: x - 1x - 12 4 Vertical asymptote: x = 1 5 Horizontal asymptote: y = 1; not intersected y x=1 6 7 41 F1x2 = Interval (؊ؕ, 1) (1, 4) (4, ؕ) Number Chosen Value of F F(0) ؍4 F(2) ؍؊2 F(5) ؍4 Location of Graph Above x-axis Below x-axis Above x-axis (2, ؊2) (5 , ) Point on Graph (0, 4) (0, 4) Q5, 14R −5 x (4, 0) (2, −2) y=1 −5 x ; Domain: 5x͉ x ≠ - 2 G is in lowest terms 3 y-intercept: 0; x-intercept: 1x + 22 4 Vertical asymptote: x = - 2 5 Horizontal asymptote: y = 0; intersected at (0, 0) ؊2 6 7 43 G1x2 = x = −2 Interval (؊ؕ, ؊2) (؊2, 0) (0, ؕ) Number Chosen ؊3 ؊1 Value of G G(؊3) ؍؊3 G(؊1) ؍؊1 Location of Graph Below x-axis Below x-axis Above x-axis Point on Graph (؊3, ؊3) (؊1, ؊1) (1 , ) G(1) ؍9 −7 y=0 (−3, −3) y Q1, 19R x (0, 0) (−1, −1) −7 AN30 Answers Section 5.3 x2 + ; Domain: 5x͉ x ≠ 2 f is in lowest terms 3 no y-intercept; no x-intercepts x 4 f is in lowest terms; vertical asymptote: x = 0 5 Oblique asymptote: y = x, not intersected 6 7 45 f 1x2 = Interval (؊ؕ, 0) (0, ؕ) Number Chosen ؊1 Value of f f(؊1) ؍؊2 Location of Graph Below x-axis Point on Graph (؊1, ؊2) x؍0 y (1, 2) y؍x x (؊1, ؊2) f (1) ؍2 Above x-axis (1, 2) 1x + 12 1x - x + 12 x3 + = ; Domain: 5x͉ x ≠ 2 f is in lowest terms 3 no y-intercept; x-intercept: - x x 4 f is in lowest terms; vertical asymptote: x = 0 5 No horizontal or oblique asymptote 6 7 y ؊1 47 f 1x2 = Interval Number Chosen (؊ؕ, ؊1) (؊1, 0) ؊2 ؊2 Value of f f (؊2) ؍ f Location of Graph Above x-axis Point on Graph (؊2, ) ؊2, (0, ؕ) ( )؍ ؊2 ؊4 f (1) ؍2 Below x-axis Above x-axis (؊ (1, 2) ) 1, ؊4 (1, 2) x (؊1, 0) ؊ ,؊ x؍0 x4 + ; Domain: 5x͉ x ≠ 2 f is in lowest terms 3 no y-intercept; no x-intercepts x3 4 f is in lowest terms; vertical asymptote: x = 0 5 Oblique asymptote: y = x, not intersected 6 49 f 1x2 = Interval (؊ؕ, 0) (0, ؕ) Number Chosen ؊1 Value of f f(؊1) ؍؊2 f(1) ؍2 Location of Graph Below x-axis Point on Graph (؊1, ؊2) 51 One possibility: R1x2 = y؍x (1, 2) x (؊1, ؊2) Above x-axis x؍0 (1, 2) x2 x2 - 53 One possibility: R1x2 = 55 (a) t-axis; C1t2 S (b) 1x - 12 1x - 32 ax2 + 1x + 12 1x - 22 57 (a) C1x2 = 16x + (b) x (c) 0.4 5000 + 100 x 12 0 (c) 0.71 h after injection b 59 (a) S 1x2 = 2x2 + (b) 40,000 x 10,000 10,000 0 y 300 0 60 (c) 2784.95 in (d) 21.54 in * 21.54 in * 21.54 in (e) To minimize the cost of materials needed for construction 63 No Each function is a quotient of polynomials, but it is not written in lowest terms Each function is undefined for x = 1; each graph has a hole at x = 1. 69 If there is a common factor between the numerator and the denominator, and the factor yields a real zero, then the graph will have a hole 17 70 4x3 - 5x2 + 2x - 2 71 e - f 72 73 ≈ - 0.164 10 (d) Approximately 17.7 ft by 56.6 ft (longer side parallel to river) 61 (a) C1r2 = 12pr + (b) 4000 r 6000 0 10 The cost is smallest when r = 3.76 cm 5.4 Assess Your Understanding (page 372) c F (a) 5x 0 x or x 6; 10, 12 ∪ 12, q (b) 5x x … or … x … 6; - q , ∪ 1, (a) 5x - x or x 6; - 1, 02 ∪ 11, q (b) 5x x - or … x 6; - q , - 12 ∪ 0, 12 5x x or x 6; - q , 02 ∪ 10, 32 11 5x͉ x … 6; - q , 13 5x x … - or x Ú 6; - q , - ∪ 2, q Answers Section 5.5 AN31 15 5x - x - or x 6; - 4, - 12 ∪ 10, q 17 5x - x … - 6; - 2, - 1] 19 5x x - 6; - q , - 22 21 5x x 6; 14, q 23 5x - x or x 6; 1- 4, 02 ∪ 10, q 25 5x x … or … x … 6; 1- q , 1] ∪ 32, 34 27 5x - x or x 6; - 1, 02 ∪ 13, q 29 5x x - or x 6; - q , - 12 ∪ 11, q 31 5x x - or x 6; - q , - 12 ∪ 11, q 33 5x x - or x 6; - q , - 12 ∪ 11, q 35 5x x … - or x … 6; - q , - 1] ∪ 10, 37 5x x - or x 6; - q , - 12 ∪ 11, q 39 5x x 6; - q , 22 41 5x - x … 6; - 2, 43 5x x or x 6; - q , 22 ∪ 13, 52 45 5x x - or - … x … - or x = or x 6; - q , - 52 ∪ 3- 4, - 34 ∪ 506 ∪ 11, q 2 47 e x ` - 1 x or x f; a - , b ∪ 13, q 2 49 5x - x or x 6; - 1, 32 ∪ 15, q 2 1 f ; - q , - 4 ∪ c , q b 53 5x x or x Ú ; - q , 32 ∪ [7, q 55 5x x 6 ; - q , 22 2 3 57 e x ` x - or x f ; a - q , - b ∪ a0, b 59 5x x … - or … x … ; - q , - ∪ 0, 3 63 (a) y 65 5x x ; 14, q y y ؍x ؉3 61 (a) 15 4, 10 10 28 67 5x x … - or x Ú ; - q , - ∪ 32, q , (؊6, 0) , 10 69 5x x - or x Ú ; - q , - 42 ∪ 32, q (؊4, 0) (0, 2) y؍1 51 e x ` x … - or x Ú ,؊ (1, 0) 10 x 10 x (؊1, 0) x؍2 x ؍؊2 (b) - 4, - 22 ∪ - 1, 32 ∪ 13, q (b) - q , - ∪ 1, 22 ∪ 12, q 71 73 y f(x) ؍x4 ؊ 2.5 (؊1, 0) (1, 0) 2.5 x y g(x) ؍3x 32 (2, 12) (؊2, 12) 2.5 x f(x) ؍x4 ؊ g(x) ؍؊2x2 ؉ f 1x2 … g 1x2 if - … x … f 1x2 … g 1x2 if - … x … Historical Problems (page 386) b b b b + b ax b + c ax b + d 3 3 b x b 2b x b bc x3 - bx2 + + bx2 + + cx + d 27 3 b 2b bc x3 + ac bx + a + db 27 b2 2b3 bc Let p = c and q = + d Then x3 + px + q 27 3 3HK = - p p K = 3H p H3 + a b = -q 3H p3 H3 = -q 27H 27H - p3 = - 27qH 27H + 27qH - p3 = ax - H3 = H3 = H3 = H = - 27q { 127q2 - 1272 - p3 2 # 27 -q -q { { -q C 2 27 q C 22 127 2 q2 C4 + Choose the positive root for now + q2 B + p3 27 + 1272p 22 127 2 = = 75 Produce at least 250 bicycles 77 (a) The stretch is less than 39 ft (b) The ledge should be at least 84 ft above the ground for a 150-lb jumper 79 At least 50 students must attend. 84 c , q b 85 x2 - x - 4 86 3x2y4 1x + 2y2 12x - 3y2 87 y = 2x 2. 1H + K2 + p 1H + K2 + q H + 3H K + 3HK2 + K3 + pH + pK + q Let 3HK H - pH - pK + K3 + pH + pK + q = 0, H + K3 = = = = 0 - p -q = = H + K3 = - q K3 = - q - H K3 = - q - J K3 = K = x = H + K x = -q -q - -q -q q2 C4 + + q q2 C4 q2 + p3 p3 27 R p3 27 C - A + + + + p3 27 -q - q2 + p3 D C4 27 D C4 27 ( Note that if we had used the negative root in 3, the result would have been the same.) x = 3 x = 2 x = p3 27 5.5 Assess Your Understanding (page 386) a f 1c2 b F 0 10 T 11 R = f 122 = 8; no 13 R = f 122 = 0; yes 15 R = f - 32 = 0; yes 17 R = f - 42 = 1; no 19 R = f a b = 0; yes 21 7; or positive; or negative 23 6; or positive; or negative 25 3; or positive; negative 27 4; or positive; or negative 29 5; positive; or negative 31 6; positive; negative AN32 Answers Section 5.5 1 1 1 33 {1, { 35 {1, {3 37 {1, {2, { , { 39 {1, {3, {9, { , { , { , { , { 2 2 5 10 20 41 {1, {2, {3, {4, {6, {12, { , { 43 {1, {2, {4, {5, {10, {20, { , { , { , { , { , { , { , { , { , { 2 2 3 3 3 6 1 45 - 3, - 1, 2; f 1x2 = 1x + 32 1x + 12 1x - 22 47 ; f 1x2 = ax - b 1x + 12 49 2, 25, - 25; f 1x2 = 1x - 22 1x - 252 1x + 252 2 1 51 - 1, , 23, - 23; f 1x2 = 1x + 12 ax - b 1x - 232 1x + 232 53 1, multiplicity 2; - 2, - 1; f 1x2 = 1x + 22 1x + 12 1x - 12 2 1 2 55 - 1, - ; f 1x2 = 1x + 12 ax + b 1x + 22 57 - 1, 59 e , - + 22, - - 22 f 61 e , 25, - 25 f 63 - 3, - 4 3 1 65 e - f 67 e , 2, f 69 LB = - 2; UB = 2 71 LB = - 1; UB = 1 73 LB = - 2; UB = 2 75 LB = - 1; UB = 77 LB = - 2; UB = 3 79 f 102 = - 1; f 112 = 10 81 f - 52 = - 58; f - 42 = 2 83 f 11.42 = - 0.17536; f 11.52 = 1.40625 85 0.21 87 - 4.04 89 1.15 91 2.53 93 95 y 30 (؊2, 4) (؊3, 0) (؊1, 0) (؊4, ؊18) 101 (1, 2) (2, 0) x (0, ؊6) y 16 (0, 2) (1, 0) x (؊2, 0) (؊1.5, ؊1.5625) ͙2 ؊ ,0 (2, 0) x (1, ؊3) (؊1, ؊9) y ؊ ,0 (1, 0) 2.5 x ,0 2.5 x (0, ؊2) (0, ؊2) ͙2 ,0 y (0, 2) 99 y 2.5 (؊1, 0) x ,0 (0, ؊1) 103 (2, 12) (؊1, 0) 97 y (3, 24) 105 - 8, - 4, - 107 k = 5 109 - 111 If f 1x2 = xn - c n, then f 1c2 = c n - c n = 0, so x - c is a factor of f 113 5 115 in. 117 All the potential rational zeros are integers, so r either is an integer or is not a rational zero (and is therefore irrational) 119 0.215 121 No; by the Rational Zeros Theorem, is not a potential rational zero. 123 No; by the Rational Zeros Theorem, is not a potential rational zero 3 124 y = x - 125 3, 82 126 10, - 232, 10, 232, 14, 02 127 - 3, 22 and 15, q 5 5.6 Assess Your Understanding (page 394) one - 4i T F + i - i, - i 11 - i, - 2i 13 - i 15 - i, - + i 17 f 1x2 = x4 - 14x3 + 77x2 - 200x + 208; a = 1 19 f 1x2 = x5 - 4x4 + 7x3 - 8x2 + 6x - 4; a = 1 21 f 1x2 = x4 - 6x3 + 10x2 - 6x + 9; a = 1 23 - 2i, 4 25 2i, - 3, 27 + 2i, - 2, 5 29 4i, - 211, 211, 23 23 23 23 i, - + i; f 1x2 = 1x - 12 ax + + i b ax + ib 31 1, - 2 2 2 2 33 2, - 2i, + 2i; f 1x2 = 1x - 22 1x - + 2i2 1x - - 2i2 35 - i, i, - 2i, 2i; f 1x2 = 1x + i2 1x - i2 1x + 2i2 1x - 2i2 1 37 - 5i, 5i, - 3, 1; f 1x2 = 1x + 5i2 1x - 5i2 1x + 32 1x - 12 39 - 4, , - 3i, + 3i; f 1x2 = 1x + 42 ax - b 1x - + 3i2 1x - - 3i2 3 22 22 22 22 22 22 22 22 i, + i, i, + i 2 2 2 2 45 Zeros that are complex numbers must occur in conjugate pairs; or a polynomial with real coefficients of odd degree must have at least one real zero 47 If the remaining zero were a complex number, its conjugate would also be a zero, creating a polynomial of degree 49 50 - 22 y 51 6x3 - 13x2 - 13x + 20 52 A = 9p ft ≈ 28.274 ft 2; C = 6p ft ≈ 18.850 ft2 41 130 43 (a) f 1x2 = 1x2 - 22x + 12 1x2 + 22x + 12 (b) - 10 x −2 −5 Review Exercises (page 397) Polynomial of degree 5 Rational Neither Polynomial of degree y y 15 (؊2, 0) (؊4, ؊8) (1, 0) (0, 8) x (0, ؊1) y 18 x (2, ؊1) (0, 3) (1, 2) (2, 3) x Answers Review Exercises AN33 y = x3 2 x-intercepts: - 4, - 2, 0; y-intercept: 3 - 4, - 2, (all multiplicity 1), crosses 4 f - 52 = - 15; f - 32 = 3; f - 12 = - 3; f 112 = 15 (؊2, 0) (؊3, 3) y 20 (؊4, 0) (؊5, ؊15) y = x3 2 x-intercepts: - 4, 2; y-intercept: 16 3 - 4, multiplicity 1, crosses; 2, multiplicity 2, touches 4 5 f - 52 = - 49; f - 22 = 32; f 132 = y (0, 16) (؊2, 32) (1, 15) (0, 0) x (؊1, ؊3) (3, 7) (2, 0) 10 x (؊4, 0) (؊5, ؊49) 10 y = - 2x3 f 1x2 = - 2x2 1x - 22 x-intercepts: 0, 2; y-intercept: 3 0, multiplicity 2, touches; 2, multiplicity 1, crosses 4 5 f - 12 = 6; f 112 = 2; f 132 = - 18 y 20 (1, 2) (؊1, 6) (2, 0) x (0, 0) ؊60 (3, ؊18) 11 y = x4 2 x-intercepts: - 3, - 1, 1; y-intercept: 3 - 3, - (both multiplicity 1), crosses; 1, multiplicity 2, touches 4 5 f - 42 = 75; f - 22 = - 9; f 122 = 15 12 Domain: 5x x ≠ - 3, x ≠ 6; horizontal asymptote: y = 0; vertical asymptotes: x = - 3, x = 13 Domain: 5x͉ x ≠ 6; oblique asymptote: y = x + 2; vertical asymptote: x = 14 Domain: 5x͉ x ≠ - ; horizontal asymptote: y = 1; vertical asymptote: x = - y 80 (؊4, 75) (2, 15) (1, 0) x (؊2, ؊9) (0, 3) (؊1, 0) (؊3, 0) 1x - 32 ; domain: 5x x ≠ 2 R is in lowest terms 3 no y-intercept; x-intercept: x 4 R is in lowest terms; vertical asymptote: x = 0 5 Horizontal asymptote: y = 2; not intersected 6 15 R1x2 = Interval (؊ؕ, 0) (0, 3) Number Chosen ؊2 Value of R R(؊2) ؍5 R(1) ؍؊4 R(4) – ؍2 Location of Graph Above x-axis Below x-axis Above x-axis Point on Graph (1, ؊4) (؊2, 5) (3, ؕ) y 10 (؊2, 5) y؍2 (3, 0) x (1, ؊4) ؊10 4, x؍0 (4, ) 1– 16 Domain: 5x x ≠ 0, x ≠ 2 H is in lowest terms 3 no y-intercept; x-intercept: - 4 H is in lowest terms; vertical asymptotes: x = 0, x = 2 5 Horizontal asymptote: y = 0; intersected at - 2, 02 x؍0 ؊2 6 y Interval (؊ؕ, ؊2) Number Chosen ؊3 Value of H H(؊3) ؍؊ –– 15 Location of Graph Below x-axis Point on Graph (؊3, ؊ ) –– 15 ؊1, (؊2, 0) (0, 2) (2, ؕ) ؊1 H(؊1) ؍1–3 H(1) ؍؊3 H(3) ؍5–3 Above x-axis Below x-axis Above x-axis (؊1, ) 1– (1, ؊3) 5 3, (؊2, 0) x ؊3, ؊ 15 (3, ) (1, ؊3) 5– x؍2 1x + 32 1x - 22 ; domain: 5x x ≠ - 2, x ≠ 2 R is in lowest terms 3 y-intercept: 1; x-intercepts: - 3, 1x - 32 1x + 22 4 R is in lowest terms; vertical asymptotes: x = - 2, x = 3 5 Horizontal asymptote: y = 1; intersected at 10, 12 6 ؊3 ؊2 y (0, 1) 17 R1x2 = Interval (؊ؕ, ؊3) (؊3, ؊2) Number Chosen ؊4 Value of R R(؊4) –– ؍ 5– R (0) ؍1 R Below x-axis Above x-axis Below x-axis (؊ (0, 1) ( –– (3, ؕ) ( ) (؊4, ) (2, 3) R ؊ –2 ؍؊–– 11 ؊ –2 Location of Graph Above x-axis Point on Graph (؊2, 2) 5– 2, ) ؊–– 11 ( ) ؍؊ 5– 5– 2, ؊4, 11 –– ) 11 ؊ –– Above x-axis –– ؊ y؍1 x (2, 0) (؊3, 0) R(4) –– ؍ (4, ) 4, 11 ,؊ ,؊ 11 x ؍؊2 x ؍3 x3 ; domain: 5x x ≠ - 2, x ≠ 2 F is in lowest terms 3 y-intercept: 0; x-intercept: 1x + 22 1x - 22 4 F is in lowest terms; vertical asymptotes: x = - 2, x = 2 5 Oblique asymptote: y = x; intersected at 10, 02 6 ؊2 27 x ؍؊2 18 F1x2 = Interval (؊ؕ, ؊2) Number Chosen ؊3 Value of F F(؊3) ؍؊–– 27 Location of Graph Below x-axis Point on Graph (؊3, ؊ ) 27 –– (؊2, 0) (0, 2) (2, ؕ) ؊1 F(؊1 ) ؍1–3 F(1) ؍؊1–3 –– F(3) ؍27 Above x-axis Below x-axis Above x-axis (؊1, ) 1– (1 , ؊ ) 1– 3 (0, 0) ؊1, (3, ) 27 – 3, y 10 x 1, ؊ 27 ؊3, ؊ y؍x x؍2 AN34 Answers Review Exercises 19 Domain: 5x x ≠ 2 R is in lowest terms 3 y-intercept: 0; x-intercept: 4 R is in lowest terms; vertical asymptote: x = 1 5 No oblique or horizontal asymptote 6 Interval (؊ؕ, 0) (0, 1) (1, ؕ) Number Chosen ؊2 1– 2 Value of R R(؊2) –– ؍ 32 R 20 G1x2 = (؊2, ) ( 32 –– 1x + 22 1x - 22 1x + 12 1x - 22 – 1– , 1– ; domain: 5x x ≠ - 1, x ≠ 2 In lowest terms, G1x2 = ؊2 Interval (؊ؕ, ؊2) (؊2, ؊1) Number Chosen ؊3 ؊ –2 Value of G G (؊3) –؍2 Location of Graph Above x-axis Point on Graph (؊3, ) 1– (2, ؕ) G (0) ؍2 G (3) – ؍4 Below x-axis Above x-axis Above x-axis ( ) 3– 2, ) ؊1 (3, ) 5– (0, 2) 21 5x x - or - x 6 ; - q , - 22 ∪ - 1, 22 22 5x - … x … - or x Ú - 4, - ∪ 1, q 24 5x … x … or x 6; 1, ∪ 13, q 25 5x x - or x or x 6; - q , - 42 ∪ 12, 42 ∪ 16, q 2 ؊2 ؊1 ؊4 ؊4 y 2, (0, 2) y؍1 x (؊2, 0) 3, ؊ , ؊1 x ؍؊1 ؊3, G ؊ –2 ؍؊1 (؊ x + 3 y-intercept: 2; x-intercept: - x + b 5 Horizontal asymptote: y = 1, not intersected ؊1 (؊1, 2) x x؍1 (2, 32) 4 Vertical asymptote: x = - 1; hole at a2, 6 1 , 2 (0, 0) Above x-axis ) (2, 32) 32 ؊2, R(2) ؍32 Above x-axis Location of Graph Above x-axis Point on Graph ( )؍ – y 40 23 5x x or x 6; - q , 12 ∪ 12, q 2 ؊1 26 R = 10; g is not a factor of f. 27 R = 0; g is a factor of f. 28 f 142 = 47,105 29 4, 2, or positive; or negative 30 positive; or negative 1 1 31 {1, {3, { , { , { , { , { , { , { 32 - 2, 1, 4; f 1x2 = 1x + 22 1x - 12 1x - 42 2 4 12 1 33 , multiplicity 2; - 2; f 1x2 = ax - b 1x + 22 34 2, multiplicity 2; f 1x2 = 1x - 22 1x2 + 52 35 - 3, 36 e - 3, - 1, - , f 2 37 lb: - 2; ub: 3 38 lb: - 3; ub: 5 39 f 102 = - 1; f 112 = 1 40 f 102 = - 1; f 112 = 1 41 1.52 42 0.93 43 - i; f 1x2 = x3 - 14x2 + 65x - 102 44 - i, - i; f 1x2 = x4 - 2x3 + 3x2 - 2x + 2 45 - 2, 1, 4; f 1x2 = 1x + 22 1x - 12 1x - 42 1 1multiplicity 22; f 1x2 = 1x + 22 ax - b 47 (multiplicity 2), - 15i, 15i; f 1x2 = 1x + 15i2 1x - 15i2 1x - 22 2 12 12 12 12 48 - 3, 2, i, i; f 1x2 = 1x + 32 1x - 22 ax + i b ax ib 2 2 46 - 2, 49 (a) A1r2 = 2pr + 50 (a) 500 r 280 (b) 223.22 cm (c) 257.08 cm2 (d) 190 1000 0 The relation appears to be cubic (b) P1t2 = 4.4926t - 45.5294t + 136.1209t + 115.4667; ≈ +928,000 (c) 280 A is smallest when r ≈ 3.41 cm 190 Answers Cumulative Review AN35 Chapter Test (page 399) (a) p 15 (b) : { , {1, { , { , {3, {5, { , {15 q 2 2 (c) - 5, - , 3; g 1x2 = 1x + 52 12x + 12 1x - 32 (d) y-intercept: - 15; x-intercepts: - 5, - , y (4, ؊1) x (3, ؊2) (2, ؊1) (e) Crosses at - 5, - , (f) y = 2x3 (g) ؊ ,0 (؊2, 45) y (؊3, 60) 60 (3, 0) x (؊5, 0) (2, ؊35) (0, ؊15) (1, ؊36) - 161 + 161 , f Domain: 5x x ≠ - 10, x ≠ ; asymptotes: x = - 10, y = 6 Domain: 5x x ≠ - ; asymptotes: x = - 1, y = x + y؍x؉1 1x - 92 1x - 12 y Answers may vary One possibility is r 1x2 = 1x - 42 1x - 92 (؊3, 0) (1, 0) 10 f 102 = 8; f 142 = - 36 x Since f 102 = and f 142 = - 36 0, the Intermediate Value Theorem guarantees that there (0, ؊3) is at least one real zero between and x ؍؊1 11 5x x or x 6; - q , 32 ∪ 18, q Answers may vary One possibility is f 1x2 = x4 - 4x3 - 2x2 + 20x 4, - 5i, 5i e 1, Cumulative Review (page 399) 5x - x 6; - 1, 42 126 5x x … or x Ú ; - q , or 1, q ؊1 f 1x2 = - 3x + y (؊1, 4) 5 y = 2x - y 6 y 10 (3, 5) (؊1, ؊1) x x (؊2, ؊8) 3 f; c , q b 2 11 x-intercepts: - 3, 0, 3; y-intercept: 0; symmetric with respect to the origin 17 12 y = - x + 13 Not a function; it fails the vertical-line test 3 14 (a) 22 (b) x2 - 5x - 2 (c) - x2 - 5x + 2 (d) 9x2 + 15x - 2 (e) 2x + h + 7 15 (a) 5x x ≠ (b) No; 12, 72 is on the graph. (c) 4; 13, 42 is on the graph. (d) ; a , b is on the graph 4 (e) Rational Not a function; has two images. 50, 2, e x ` x Ú 10 Center: - 2, 12 ; radius: (؊2, 4) y (؊5, 1) (؊2, 1) (؊2, ؊2) 16 (1, 1) x 17 y (0, 7) y ,0 (0, 1) ͙2 1؊ ,0 x 21 (a) Domain: 5x x - or - 3, q (b) x-intercept: - ; y-intercept: y (c) ؊ ,0 (2, 8) (1, 1) 10 x (2, 5) (0, 1) x (2, ؊2) (؊3, ؊5) (d) Range: 5y y 6 or - q , 52 ͙2 1؉ ,0 x (1, ؊1) x؍1 22 y (؊1, 5) (؊2, 2) (0, 2) x 18 6; y = 6x - 19 (a) x-intercepts: - 5, - 1, 5; y-intercept: - (b) No symmetry (c) Neither (d) Increasing: - q , - 32 and 12, q ; decreasing: - 3, 22 (e) A local maximum value of occurs at x = - (f) A local minimum value of - occurs at x = 20 Odd 23 (a) 1f + g2 1x2 = x2 - 9x - 6; domain: all real numbers f x2 - 5x + (b) a b 1x2 = ; domain: e x ` x ≠ - f g - 4x - 24 (a) R1x2 = x + 150x 10 (b) +14,000 (c) 750, +56,250 (d) +75 AN36 Answers Section 6.1 Chapter 6 Exponential and Logarithmic Functions 6.1 Assess Your Understanding (page 408) composite function; f(g(x)) F c a F (a) - 1 (b) - 1 (c) 8 (d) 0 (e) 8 (f) - 7 11 (a) 4 (b) 5 (c) - 1 (d) - 163 1 13 (a) 98 (b) 49 (c) 4 (d) 4 15 (a) 97 (b) (c) 1 (d) - 17 (a) 22 (b) 22 (c) 1 (d) 0 19 (a) (b) (c) 1 (d) 2 17 21 (a) (b) 1 (c) (d) 0 23 (a) (f ∘ g)(x) = 6x + 3; all real numbers (b) (g ∘ f)(x) = 6x + 9; all real numbers + (c) (f ∘ f)(x) = 4x + 9; all real numbers (d) (g ∘ g)(x) = 9x; all real numbers 25 (a) (f ∘ g)(x) = 3x2 + 1; all real numbers (b) (g ∘ f)(x) = 9x2 + 6x + 1; all real numbers (c) (f ∘ f)(x) = 9x + 4; all real numbers (d) (g ∘ g)(x) = x4; all real numbers 27 (a) (f ∘ g)(x) = x4 + 8x2 + 16; all real numbers (b) (g ∘ f)(x) = x4 + 4; all real numbers (c) (f ∘ f)(x) = x4; all real numbers 2(x - 1) 3x (d) (g ∘ g)(x) = x4 + 8x2 + 20; all real numbers 29 (a) (f ∘ g)(x) = ; 5x͉ x ≠ 0, x ≠ (b) (g ∘ f)(x) = ; 5x͉ x ≠ - x 3(x - 1) (c) (f ∘ f)(x) = ; 5x͉ x ≠ 1, x ≠ (d) (g ∘ g)(x) = x; 5x͉ x ≠ 31 (a) (f ∘ g)(x) = ; 5x͉ x ≠ - 4, x ≠ - x + x - 4(x - 1) (b) (g ∘ f)(x) = ; 5x͉ x ≠ 0, x ≠ (c) (f ∘ f)(x) = x; 5x͉ x ≠ (d) (g ∘ g)(x) = x; 5x͉ x ≠ x 33 (a) (f ∘ g)(x) = 22x + 3; e x͉ x Ú - f (b) (g ∘ f)(x) = 2x + 3; 5x͉ x Ú (c) (f ∘ f)(x) = x; 5x͉ x Ú (d) (g ∘ g)(x) = 4x + 9; all real numbers 35 (a) (f ∘ g)(x) = x; 5x͉ x Ú (b) (g ∘ f)(x) = ͉ x ͉ ; all real numbers 4x - 17 ; e x ` x ≠ 3; x ≠ f 2x - 2x + 3x - 11 5x͉ x ≠ - 4; x ≠ - (c) (f ∘ f)(x) = ; 5x͉ x ≠ - 1; x ≠ (d) (g ∘ g)(x) = ; ex ` x ≠ ; x ≠ 3f x - 2x - 11 1 f a x b = a x b = x; (g ∘ f )(x) = g( f (x)) = g(2x) = (2x) = x 2 (c) (f ∘ f)(x) = x4 + 2x2 + 2; all real numbers (d) (g ∘ g)(x) = 2x - - 1; 5x͉ x Ú 37 (a) (f ∘ g)(x) = (b) (g ∘ f)(x) = - 3x - ; 2x + 39 ( f ∘ g)(x) = f(g(x)) = 3 3 41 (f ∘ g)(x) = f(g(x)) = f( x) = ( x)3 = x; (g ∘ f)(x) = g(f(x)) = g(x3) = x = x 1 1 43 ( f ∘ g)(x) = f(g(x)) = f a (x + 6) b = c (x + 6) d - = x + - = x; (g ∘ f)(x) = g(f(x)) = g(2x - 6) = (2x - + 6) = (2x) = x 2 2 1 45 ( f ∘ g)(x) = f(g(x)) = f a (x - b) b = a c (x - b) d + b = x; (g ∘ f)(x) = g(f (x)) = g(ax + b) = (ax + b - b) = x a a a 47 f(x) = x4; g(x) = 2x + (Other answers are possible.) 49 f(x) = 2x; g(x) = x2 + (Other answers are possible.) 51 f(x) = ͉ x͉ ; g(x) = 2x + (Other answers are possible.) 53 (f ∘ g)(x) = 11; (g ∘ f)(x) = 2 55 - 3, 3 57 (a) (f ∘ g)(x) = acx + ad + b (b) (g ∘ f)(x) = acx + bc + d (c) The domains of both f ∘ g and g ∘ f are all real numbers. (d) f ∘ g = g ∘ f when ad + b = bc + d 2100 - p 16 pt 61 C(t) = 15,000 + 800,000t - 40,000t 2 63 C(p) = + 600, … p … 100 65 V(r) = 2pr 25 67 (a) f(x) = 0.7235x (b) g(x) = 141.119x (c) g(f(x)) = g(0.7235x) = 102.0995965x (d) 102,099.5965 yen 69 (a) f(p) = p - 200 (b) g(p) = 0.8p (c) (f ∘ g)(p) = 0.8p - 200; (g ∘ f)(p) = 0.8p - 160; The 20% discount followed by the $200 rebate is the better deal. 71 15 73 f is an odd function, so f( - x) = - f(x) g is an even function, so g( - x) = g(x) Then (f ∘ g)( - x) = f(g( - x)) = f(g(x)) = (f ∘ g)(x) So f ∘ g is even Also, (g ∘ f)( - x) = g(f( - x)) = g( - f(x)) = g(f(x)) = (g ∘ f)(x), so g ∘ f is even 59 S(t) = 74 (f + g)(x) = 4x + 3; Domain: all real numbers (f - g)(x) = 2x + 13; Domain: all real numbers (f ∘ g)(x) = 3x2 - 7x - 40; Domain: all real numbers f 3x + ¢ ≤(x) = ; Domain: 5x ͉ x ≠ g x - 75 , 4 76 10 Ϫ3 77 Domain: 5x ͉ x ≠ Vertical asymptote: x = Oblique asymptote: y = x + Ϫ10 Local minimum: - 5.08 at x = - 1.15 Local maximum: 1.08 at x = 1.15 Decreasing: ( - 3, - 1.15); (1.15, 3) Increasing: ( - 1.15, 1.15) 6.2 Assess Your Understanding (page 419) f(x1) ≠ f(x2) one-to-one 3 y = x [4, q 10 T 11 a 12 d 13 one-to-one 15 not one-to-one 17 not one-to-one 19 one-to-one 21 one-to-one 23 not one-to-one 25 one-to-one Answers Section 6.2 AN37 27 29 Annual Rainfall (inches) Monthly Cost of Life Insurance Location Atlanta, Georgia Boston, Massachusetts Las Vegas, Nevada Miami, Florida Los Angeles, California 49.7 43.8 4.2 61.9 12.8 Age $10.59 30 40 45 $12.52 $15.94 Domain: {+10.59, +12.52, +15.94} Range: {30, 40, 45} Domain: {49.7, 43.8, 4.2, 61.9, 12.8} Range: {Atlanta, Boston, Las Vegas, Miami, Los Angeles} 31 15, - 32, 19, - 22, 12, - 12, 111, 02, - 5, 12 Domain: 55, 9, 2, 11, - Range: - 3, - 2, - 1, 0, 1 35 f 1g 1x2 = f a 1x - 42 b = c 1x - 42 d + = (x - 4) + = x 3 1 g 1f 1x2 = g 13x + 42 = [ 13x + 42 - 4] = 13x2 = x 3 33 11, - 22, 12, - 32, 10, - 102, 19, 12, 14, 22 Domain: 51, 2, 0, 9, Range: - 2, - 3, - 10, 1, 3 39 f 1g 1x2 = f x + 82 = x + 82 - = 1x + 82 - = x 3 3 g 1f 1x2 = g 1x - 82 = 1x - 82 + = 2x = x 4x - b - x 4x - + - x 14x - 32 + 12 - x2 = = 4x - + 12 - x2 4x - 43 f 1g 1x2 = f a b = - x 2a + 5x = x, x ≠ 1 f(f -1(x)) = f a x b = a x b = x 3 -1 -1 f (f(x)) = f (3x) = (3x) = x (b) Domain of f = Range of f -1 = All real numbers; Range of f = Domain of f -1 = All real numbers (c) y؍x y f(x) ؍3x f 47 y؍x y 2.5 (1, 2) (2, 1) 49 y؍x y؍x y 2.5 f ؊1 2.5 x (؊2, ؊2) f ؊1 2.5 x (؊1, ؊1) (1, 0) f ؊1 2.5 x f ؊1 (0, ؊1) x x x f(f -1(x)) = f a - b = a - b + 4 = (x - 2) + = x 4x + f -1(f(x)) = f -1(4x + 2) = 1 = ax + b = x 2 (b) Domain of f = Range of f -1 = All real numbers; 53 (a) f -1 1x2 = Range of f = Domain of f -1 = All real numbers y؍x y (c) x ؊1 y 2.5 1 = x; x ≠ 0, g(f (x)) = g a b = = x, x ≠ x 1 a b a b x x 4a x 51 (a) f -1(x) = 41 f(g(x)) = f a b = x 2x + b - x + 2x + x + 4 12x + 32 - 1x + 42 5x = = = x, x ≠ - 1x + 42 - 12x + 32 2x + g 1f 1x2 = g a b = x + 45 x x + b = c + d - = 1x + 82 - = x 4 4x - g 1f 1x2 = g 14x - 82 = + = 1x - 22 + = x 37 f 1g 1x2 = f a (x) ؍x f(x) ؍4x ؉ x f ؊1(x) ؍ 55 (a) f -1 1x2 = 2x + f 1f -1 1x2 = f x + 12 = 2x + 12 - = x f -1 1f 1x2 = f -1 1x3 - 12 = 1x3 - 12 + = x (b) Domain of f = Range of f -1 = All real numbers; Range of f = Domain of f -1 = All real numbers y؍x y (c) 57 (a) f -1(x) = 2x - 4, x Ú f(f -1(x)) = f( 2x - 4) = ( 2x - 4)2 + = x f -1(f(x)) = f -1(x2 + 4) = 2(x2 + 4) - = 2x2 = x, x Ú (b) Domain of f = Range of f -1 = 5x ͉ x Ú 6; Range of f = Domain of f -1 = 5x ͉ x Ú (c) f (x) ؍x ؉ 4, y xՆ0 y؍x f ؊1(x) ͙ ؍x ؉ f x f(x) ؍x ؊ 1 x ؊ (x) ͙ ؍x ؊ ؊1 x AN38 Answers Section 6.2 59 (a) f -1(x) = 2x + x 2x + 1 x f 1f -1 1x2 = f a b = = = x x 2x + 12x + 12 - 2x - x 2a b + + 1x - 22 x - -1 -1 f 1f 1x2 = f a b = = = x x - 1 x - (b) Domain of f = Range of f -1 = 5x ͉ x ≠ 6; Range of f = Domain of f -1 = 5x ͉ x ≠ (c) x؍0 x f(f -1(x)) = f a b = x 4 a b x f -1(f(x)) = f -1 a b = x 61 (a) f -1 1x2 = = x 4 a b x = x (b) Domain of f = Range of f -1 = 5x ͉ x ≠ 6; Range of f = Domain of f -1 = 5x ͉ x ≠ (c) y؍x y f ؊1(x) ؍ f (x) ؍f ؊1(x) ؍x 2x ؉ x y x؍2 y؍x y؍2 y؍0 x x f(x) ؍ - 3x x - 3x f 1f -1 1x2 = f a b = x 63 (a) f -1 1x2 = 2x 2x = = = x - 3x 3x + - 3x + x 2 - 3a b 13 + x2 - # + x 2x f -1 1f 1x2 = f -1 a b = = = = x + x 2 + x (b) Domain of f = Range of f -1 = 5x ͉ x ≠ - 6; Range of f = Domain of f -1 = 5x ͉ x ≠ - 2x 65 (a) f -1 1x2 = x - - 2x 3a b - 2x2 x - - 2x - 6x -1 f 1f 1x2 = f a b = = = = x x - - 2x - 2x + 1x - 32 -6 + x - 3x - 2a b - 13x2 x + 3x - 6x -1 -1 f 1f 1x2 = f a b = = = = x x + 3x 3x - 1x + 22 -6 - x + (b) Domain of f = Range of f -1 = 5x ͉ x ≠ - 6; Range of f = Domain of f -1 = 5x ͉ x ≠ x 67 (a) f -1 1x2 = 3x - x 2a b 3x - x 2x 2x f 1f -1 1x2 = f a b = = = x = 3x - 3x - 13x - 22 x 3a b - 3x - 2x f -1 1f 1x2 = f -1 a b = 3x - 2x 3x - 2x 3a b - 3x - (b) Domain of f = Range of f -1 = e x ͉ x ≠ 69 (a) f -1 1x2 = 3x + 2x - 3x + f 1f -1 1x2 = f a b = 2x - 3a 3x + b + 2x - 3a = 3x + b + 2x - 3x + 2a b - 2x - (b) Domain of f = Range of f -1 = e x ͉ x ≠ 2x 2x = = x 6x - 13x - 12 2 f ; Range of f = Domain of f -1 = e x ` x ≠ f 3 3x + 2a b - 2x - 3x + f -1 1f 1x2 = f -1 a b = 2x - = 13x + 42 + 12x - 32 13x + 42 - 12x - 32 = = 13x + 42 + 12x - 32 13x + 42 - 12x - 32 17x = x 17 = 17x = x 17 3 f ; Range of f = Domain of f -1 = e x ` x ≠ f 2 x؊2 Answers Section 6.3 AN39 71 (a) f -1 1x2 = f 1f -1 - 2x + x - - 2x + 1x2 = f a b = x - 2x + f -1 1f 1x2 = f -1 a b = x + (b) Domain of f = Range of f -1 73 (a) f -1 1x2 = 21 - 2x f 1f -1 - 2x + b + - 2x + 32 + 1x - 22 x - -x = = = x - 2x + - 2x + + 1x - 22 -1 + x - 2a 2x + b + - 12x + 32 + 1x + 22 x + -x = = = x 2x + - 1x + 22 -1 2x + - x + = {x ͉ x ≠ - 2}; Range of f = Domain of f -1 = 5x ͉ x ≠ - 2a - 4 - 11 - 2x2 - 2x 8x 1x2 = f a b = = = = x # 4 21 - 2x 2# - 2x f -1 1f 1x2 = f -1 a x2 - 2x2 b = B - 2a x - 2x2 = b A x2 = 2x2 = x, since x (b) Domain of f = Range of f -1 = 5x ͉ x 6; Range of f = Domain of f -1 = e x ͉ x f 75 (a) 0 (b) 2 (c) 0 (d) 1 77 7 79 Domain of f -1: [ - 2, q 2; range of f -1: [5, q 81 Domain of g -1: [0, q 2; range of g -1: ( - q , 0] 83 Increasing on the interval (f(0), f(5)) 85 f -1 1x2 = 1x - b2, m ≠ 0 87 Quadrant I m -1 89 Possible answer: f 1x2 = ͉ x ͉ , x Ú 0, is one-to-one; f 1x2 = x, x Ú d + 90.39 93 (a) 77.6 kg 6.97 W - 50 W + 88 + 60 = (b) h 1W2 = 6.97r - 90.39 + 90.39 6.97r 2.3 2.3 (b) r 1d 1r2 = = = r 6.97 6.97 50 + 2.3 1h - 602 + 88 2.3h (c) h 1W 1h2 = = = h d + 90.39 2.3 2.3 d 1r 1d2 = 6.97 a b - 90.39 = d + 90.39 - 90.39 = d 6.97 W + 88 W1h 1W2 = 50 + 2.3 a - 60 b (c) 56 miles per hour 2.3 = 50 + W + 88 - 138 = W (d) 73 inches 91 (a) r 1d2 = 97 (a) t represents time, so t Ú H - 100 100 - H (b) t 1H2 = = A - 4.9 A 4.9 (c) 2.02 seconds - dx + b 99 f -1 1x2 = ; f = f -1 if a = - d 103 No cx - a 95 (a) {g ͉ 36,900 … g … 89,350} (b) {T͉ 5081.25 … T … 18,193.75} T - 5081.25 (c) g 1T2 = + 36,900 0.25 Domain: 5T͉ 5081.25 … T … 18,193.75 Range: 5g ͉ 36,900 … g … 89,350 107 6xh + 3h2 - 7h 108 y −4 109 Zeros: - - 213 - + 213 - - 213 - + 213 , , x-intercepts: , 6 6 3 , x ≠ f ; Vertical asymptote: x = - , 2 Horizontal asymptote: y = 110 Domain: e x ` x ≠ - x −5 6.3 Assess Your Understanding (page 434) Exponential function; growth factor; initial value a T T 10 a - 1, b ; (0, 1); (1, a) 11 4 12 F 13 b 14 c a 15 (a) 8.815 (b) 8.821 (c) 8.824 (d) 8.825 17 (a) 21.217 (b) 22.217 (c) 22.440 (d) 22.459 19 1.265 21 0.347 23 3.320 25 149.952 27 Neither 29 Exponential; H1x2 = 4x 31 Exponential; f 1x2 = 3(2x) 33 Linear; H1x2 = 2x + 4 35 B 37 D 39 A 41 E 43 45 y ؊1, (1, 3) (0, 2) y 47 (2, 3) (1, 1) y؍0 2.5 x y؍1 2.5 x Domain: All real numbers Range: 5y ͉ y or 11, q Horizontal asymptote: y = 0, Domain: All real numbers Range: 5y ͉ y or 10, q Horizontal asymptote: y = (؊1, 6) 49 y (0, 3) 1, 2.5 x y 10 (؊1, 1) y؍0 (0, ؊1) 2.5 y ؍؊2 1, ؊ Domain: All real numbers Range: 5y ͉ y or 10, q Horizontal asymptote: y = x y؍0 Domain: All real numbers Range: 5y ͉ y - or - 2, q Horizontal asymptote: y = - AN40 51 Answers Section 6.3 53 y 33 ؊1, 16 (2, 6) ؊2, (1, 3) y؍2 x 55 y (؊2, e2) (0, 3) (2, 5) (؊1, e) (0, 1) y؍0 y؍2 2.5 x Domain: All real numbers Range: 5y ͉ y or 12, q Horizontal asymptote: y = Domain: All real numbers Range: 5y ͉ y or 12, q Horizontal asymptote: y = 59 61 y؍5 y y؍2 x x Domain: All real numbers Range: 5y ͉ y 6 or - q , 52 Horizontal asymptote: y = 101 103 y (؊2, e 2) (؊1, e) (2, e 2) ؊2, ؊ (1, e) (0, 1) x Domain: - q , q Range: [1, q Intercept: (0, 1) Domain: All real numbers Range: 5y ͉ y 6 or - q , 22 Horizontal asymptote: y = e2 y ؊1, ؊ e (0, ؊1) 1, ؊ e y؍0 x 2, ؊ e Domain: - q , q Range: [ - 1, 02 Intercept: 10, - 12 1, e y (؊1, e) x Domain: All real numbers Range: 5y ͉ y or 10, q Horizontal asymptote: y = ؊3, e (؊2, 1) y؍05 x Domain: All real numbers Range: 5y ͉ y or 10, q Horizontal asymptote: y = - 22, 0, 22 73 56 75 - 1, 77 - 4, 79 - 81 51, 83 49 85 87 5 89 f 1x2 = 3x 91 f 1x2 = - x 93 f(x) = 3x + 9 95 (a) 16; (4, 16) (b) - 4; a - 4, b 97 (a) ; a - 1, b (b) 3; (3, 66) 16 4 99 (a) 60; - 6, 602 (b) - 4; - 4, 122 (c) - 105 (a) 74% (b) 47% (c) Each pane allows only 97% of light to pass through 107 (a) $16,231 (b) $8626 (c) As each year passes, the sedan is worth 90% of its value the previous year. 109 (a) 30% (b) 9% (c) Each year only 30% of the previous survivors survive again. 111 3.35 mg; 0.45 mg 113 (a) 0.632 (b) 0.982 (c) (d) (e) About y = 1−e−0.1t 40 0 115 (a) 0.0516 (b) 0.0888 117 (a) 70.95% (b) 72.62% (c) 100% 119 (a) 5.41 amp, 7.59 amp, 10.38 amp (b) 12 amp (d) 3.34 amp, 5.31 amp, 9.44 amp (e) 24 amp (c), (f) I 30 (0, e2) 63 53 65 - 67 52 69 e f 71 y (0, 1) (0, 4) 57 y 121 36 123 Final Denominator Value of Expression Compare Value to e ? 2.718281828 + 2.5 2.5 e I2 (t ) ؍24(1 ؊ e؊0.5t ) + 2.8 2.8 e I1 (t) ؍12(1 ؊ e؊2 t ) + 2.7 2.7 e + 2.721649485 2.721649485 e + 2.717770035 2.717770035 e + 2.718348855 2.718348855 e t 125 f (A + B) = aA + B = aA # aB = f (A) # f (B) 127 f (ax) = aax = (ax)a = f (x) a (b) 129 (a) f ( - x) = (e -x + e -(-x)) (c) (cosh x)2 - (sinh x)2 2 1 -x = c (e x + e -x) d - c (e x - e -x) d x = (e + e ) 2 y = (e x + e−x ) 2 2x -2x = e + + e - e 2x + - e -2x = (e x + e -x) Ϫ6 Ϫ1 = f(x) = (4) = x -x -1 x 131 59 minutes 135 a = (a ) = ¢ ≤ 136 ( - q , - h - 2, 137 (2, q ) 138 f(x) = - 2x2 + 12x - 13 a 139 (a) y (؊3, 0) (؊2, ؊3) (1, 0) x (0, ؊3) (b) Domain: ( - q , q ); Range: - 4, q ) (c) Decreasing: ( - q , - 1); Increasing: ( - 1, q ) (؊1, ؊4) x ؍؊1 6.4 Assess Your Understanding (page 448) 5x ͉ x or 10, q a , - b , 11, 02, 1a, 12 1 F T a 10 c 11 = log 9 13 = log a 1.6 15 x = log 7.2 17 x = ln a 1 x 19 = 8 21 a = 3 23 = 2 25 e x = 4 27 0 29 2 31 - 4 33 35 4 37 39 5x ͉ x 6; 13, q 2 Answers Section 6.4 AN41 41 All real numbers except 0; 5x ͉ x ≠ ; ( - q , 0) h (0, q ) 43 5x ͉ x 10 6; 110, q 45 5x ͉ x - 6; - 1, q 47 5x ͉ x - or x 6; - q , - 12 h 10, q 49 5x ͉ x Ú 6; [1, q 51 0.511 53 30.099 55 2.303 57 - 53.991 59 22 61 y f (x) ؍3 x y؍x (1, 3) (3, 1) ؊1, (0, 1) f ؊1 (x) ؍log x 63 x (1, 0) , ؊1 73 (a) Domain: - 4, q (b) y 65 B 67 D 69 A 71 E x f (x) ؍ y (؊1, 2) ,1 y؍x 2 1, x f ؊1 (x) ؍log 1/2 x (2, ؊1) 75 (a) Domain: 10, q (b) y 77 (a) Domain: 10, q (b) y (؊3, 0) x x؍0 (c) Range: - q , q Vertical asymptote: x = (d) f -1(x) = e x - (e) Domain of f -1: - q , q Range of f -1: (0, q ) (f) y 5 x (0, ؊3) y ؍؊4 79 (a) Domain: 14, q (b) y x ؍4 2.5 (2, 1) 5, x (2, 5) y؍4 x x ؍؊3 (c) Range: - q , q Vertical asymptote: x = - (d) f -1(x) = 3x - - (e) Domain of f -1: - q , q Range of f -1: ( - 2, q ) y (f) (4, 1) y ؍؊2 x (3, ؊1) x 87 (a) Domain: - q , q (b) y (6, 8) (0, 5) y؍4 89 59 91 e f 93 52 95 55 97 53 ln 10 ln - 99 52 101 e f 103 e f 105 x (c) Range: 14, q Horizontal asymptote: y = (d) f -1(x) = log 2(x - 4) (e) Domain of f -1: 14, q Range of f -1: ( - q , q ) (f) y x x 1, (8, 6) (؊2, ؊2) (1, 4) x y ؍؊3 (c) Range: - 3, q Horizontal asymptote: y = - (d) f -1(x) = ln(x + 3) - (e) Domain of f -1: - 3, q Range of f -1: ( - q , q ) y (f) y؍0 (؊2, ؊2) (؊1, 3) (c) Range: - q , q Vertical asymptote: x = # 2x (d) f -1(x) = 10 (e) Domain of f -1: - q , q Range of f -1: (0, q ) (f) y 0, x x؍0 x 85 (a) Domain: - q , q (b) y y؍0 83 (a) Domain: - 2, q (b) x ؍؊2 y 1, (c) Range: - q , q Vertical asymptote: x = (d) f -1(x) = 10x - + (e) Domain of f -1: - q , q Range of f -1: (4, q ) (f) y ؊3, x 81 (a) Domain: 10, q (b) y 2.5 (c) Range: - q , q Vertical asymptote: x = (d) f -1(x) = e x + (e) Domain of f -1:1 - q , q Range of f -1: (0, q ) y (f) y؍0 (5, 2) x 1, ؊3 x؍0 x ؍؊4 (c) Range: - q , q Vertical asymptote: x = - (d) f -1(x) = e x - (e) Domain of f -1: ( - q , q ) Range of f -1: ( - 4, q ) (f) y (1, 2) x (5, 0) 10 x x؍4 - 22, 22 107 - f 111 e - log f 1 113 (a) e x ` x - f ; a - , q b 2 (b) 2; (40, 2) (c) 121; (121, 3) (d) 109 e ln AN42 117 y 119 (a) 1 (b) 2 (c) (d) It increases. (e) 0.000316 (f) 3.981 * 10-8 121 (a) 5.97 km (b) 0.90 km 123 (a) 6.93 min (b) 16.09 125 h ≈ 2.29, so the time between injections is about h, 17 y 2.5 (1, 0) (1, 0) x x (؊1, 0) x؍0 x؍0 Domain: 5x ͉ x ≠ Range: - q , q Intercepts: - 1, 02, (1, 0) Domain: 5x ͉ x Range: 5y ͉ y Ú Intercept: (1, 0) 127 0.2695 s 0.8959 s y Amperes 115 Answers Section 6.4 2.0 1.6 1.2 0.8 0.4 (0.8959, 1) (0.2695, 0.5) 0.4 1.2 2.0 Seconds x 129 50 decibels (dB) 131 90 dB 133 8.1 135 (a) k ? 11.216 (b) 6.73 (c) 0.41% (d) 0.14% 1 1 137 Because y = log x means 1y = = x, which cannot be true for x ≠ 1 139 Zeros: - 3, - , , 3; x-intercepts: - 3, - , , 2 2 140 12 141 f(1) = - 5; f (2) = 17 142 + i; f(x) = x4 - 7x3 + 14x2 + 2x - 20; a = 6.5 Assess Your Understanding (page 459) 0 M r log a M; log a N log a M; log a N r log a M 7 F F 10 F 11 b 12 b 13 71 15 - 4 17 7 19 1 21 23 3 25 27 4 29 a + b 31 b - a 33 3a 35 1a + b2 37 + log x 39 log z 41 + ln x 43 ln x - x 45 log a u + log a v 1 47 ln x + ln 11 - x2 49 log x - log 1x - 32 51 log x + log 1x + 22 - log 1x + 32 53 ln 1x - 22 + ln 1x + 12 - ln 1x + 42 3 1 x - 55 ln + ln x + ln 11 + 3x2 - ln 1x - 42 57 log u3v4 59 log a 5/2 b 61 log c d 63 - ln 1x - 12 65 log 2[x 13x - 22 4] 1x + 12 x 25x6 67 log a a 22x + log x 79 y = log b 69 log c 1x + 12 1x + 32 1x - 12 81 y = d 71 2.771 73 - 3.880 75 5.615 77 0.874 log 1x + 22 log 83 y = 3 Ϫ1 Ϫ2 Ϫ3 85 (a) 1f ∘ g2 1x2 = x; 5x ͉ x is any real number or - q , q (b) 1g ∘ f2 1x2 = x; 5x ͉ x or 10, q (c) (d) 1f ∘ h2 1x2 = ln x2; 5x ͉ x ≠ or ( - q , 0) h (0, q ) (e) 95 y = Ϫ2 C12x + 12 1>6 1x + 42 1>9 log 1x + 12 log 1x - 12 Ϫ4 87 y = Cx 89 y = Cx 1x + 12 91 y = Ce 3x 93 y = Ce -4x + 97 3 99 101 log a 1x + 2x2 - 12 + log a 1x - 2x2 - 12 = log a 1x + 2x2 - 12 1x - 2x2 - 12 = log a[x2 - 1x2 - 12] = log a = 103 ln 11 + e 2x = ln[e 2x 1e -2x + 12] = ln e 2x + ln 1e -2x + 12 = 2x + ln 11 + e -2x -y 105 y = f 1x2 = log a x; ay = x implies ay = a b = x, so - y = log 1>a x = - f 1x2 a 1 107 f 1x2 = log a x; f a b = log a = log a - log a x = - f 1x2 x x M -1 -1 109 log a = log a 1M # N -1 = log a M + log a N -1 = log a M - log a N, since aloga N = N -1 implies a -loga N = N; that is, log a N = - log a N -1 N - - 221 - + 221 115 { - 1.78, 1.29, 3.49} 116 A repeated real solution (double root) 117 - 2, , , 2 y 118 10 −10 10 x −10 Domain: 5x͉ x … or ( - q , 2] Range: 5y͉ y Ú or [0, q ) 6.6 Assess Your Understanding (page 465) 16 21 f 56 11 516 13 e f 15 53 17 55 19 e f 21 - 6 23 - 25 - + 21 + e ≈ 56.456 - + 25 27 e f ≈ 50.854 29 52 31 e f 33 57 35 - + 22 37 - 23, 23 39 e , 729 f 41 58 2 516 e Answers Section 6.8 AN43 ln ln 10 ln 1.2 43 5log 210 = e f ≈ 53.322 45 - log 81.2 = e f ≈ - 0.088 47 e log f = d t ≈ 50.226 ln ln ln 49 e ln ln ln p ln f ≈ 50.307 51 e f ≈ 51.356 53 50 55 e f ≈ 50.534 57 e f ≈ 51.585 ln + ln ln 0.6 + ln + ln p ln 59 50 61 e log - + 27 f ≈ - 0.315 63 5log ≈ 50.861 65 No real solution 67 5log ≈ 51.161 69 {2.79} 71 - 0.57 73 - 0.70 75 {0.57} 77 {0.39, 1.00} 79 {1.32} 81 {1.31} 83 {1} 85 516 87 e - 1, 91 ln 12 97 (a), (b) + 252 ≈ 51.444 93 e e ln # ln f 89 50 f ≈ 51.921 95 (a) 55 ; (5, 3) (b) 55 ; (5, 4) (c) 51 6; yes, at (1, 2) (d) 55 (e) e - ln 15 99 (a), (b), (c) y f (x) ؍3x ؉ 18 y 18 101 (a), (b), (c) f (x) ؍3x y g(x) ؍2؊x ؉ x؉2 g(x) ؍2 g(x) ؍10 (0.710, 6.541) 2.5 x (c) 5x ͉ x 0.710 or 10.710, q 103 (a) f (x) ؍2x ؊ x y ؍؊4 , 2͙2 x x 110 e - 3, , f 111 one-to-one 105 (a) 2047 (b) 2059 107 (a) After 4.2 yr (b) After 6.5 yr (c) After 12.8 yr y f(x) ؍2x ؉ (log3 10, 10) f 11 x + ; 5x ͉ x ≠ 3, x ≠ 11 - x + 11 113 5x ͉ x Ú 6, or 31, q ) 112 (f ∘ g)(x) = (b) 2 (c) 5x ͉ x 6 or - q , 22 6.7 Assess Your Understanding (page 474) principal I; Prt; simple interest 4 effective rate of interest $108.29 $609.50 11 $697.09 13 $1246.08 15 $88.72 17 $860.72 19 $554.09 21 $59.71 23 5.095% 25 5.127% 27 , compounded annually 29 9% compounded monthly 31 25.992% 33 24.573% 35 (a) About 8.69 yr (b) About 8.66 yr 37 6.823% 39 10.15 yr; 10.14 yr 41 15.27 yr or 15 yr, mo 43 $104,335 45 $12,910.62 47 About $30.17 per share or $3017 49 Not quite Jim will have $1057.60 The second bank gives a better deal, since Jim will have $1060.62 after yr 51 Will has $11,632.73; Henry has $10,947.89. 53 (a) $63,449 (b) $44,267 55 About $1019 billion; about $232 billion 57 $940.90 59 2.53% 61 34.31 yr 63 (a) $3686.45 (b) $3678.79 65 $6439.28 67 (a) 11.90 yr (b) 22.11 yr (c) mP = Pa1 + m = a1 + r nt b n ln m = ln a1 + t = r nt b n 69 (a) 1.59% (b) In 2029 or after 21 yr 71 22.7 yr 76 R = 0; yes 2x 77 f -1(x) = 78 - 2, 5; f(x) = (x + 2)2 (x - 5) (x2 + 1) 79 56 x - r nt r b = nt ln a1 + b n n ln m n ln a1 + 6.8 Assess Your Understanding (page 486) r b n (a) 500 insects (b) 0.02 = 2, per day (c) About 611 insects (d) After about 23.5 days (e) After about 34.7 days (a) - 0.0244 = - 2.44, per year (b) About 391.7 g (c) After about 9.1 yr (d) 28.4 yr (a) N(t) = N0e kt (b) 5832 (c) 3.9 days (a) N(t) = N0e kt (b) 25,198 9.797 g 11 9953 yr ago 13 (a) 5:18 pm (b) About 14.3 min (c) The temperature of the pizza approaches 70°F 15 18.63°C; 25.1°C 17 1.7 ppm; 7.17 days, or 172 hr 19 0.26 M; 6.58 hr, or 395 min 21 26.6 days 23 (a) In 1984, 91.8% of households did 25 (a) 27 (a) 9.23 * 10-3, or about 120 not own a personal computer (b) 0.81, or about (b) (c) 5.01, or about 100 (d) 57.91°, 43.99°, 30.07° 0 100 (b) 0.78, or 78% (c) 50 people 0 (c) 70.6% (d) During 2011 (d) As n increases, the probability decreases 3 29 (a) P(t) = 50(3)t>20 (b) 661 (c) In 48 days (d) P(t) = 50e 0.055t 30 f(x) = - x + 7 31 Neither 32 ln x + ln y - ln z 33 2 2 0 40 100 ... Locate Local Maxima and Local Minima • Use a Graph to Locate the Absolute Maximum and the Absolute Minimum • Use a Graphing Utility to Approximate Local Maxima and Local Minima and to Determine Where... Kathleen Miranda, SUNY at Old Westbury Chris Mirbaha, The Community College of Baltimore County Val Mohanakumar, Hillsborough Community College Thomas Monaghan, Naperville North High School Miguel Montanez,... Chicago Susan Sandmeyer, Jamestown Community College Brenda Santistevan, Salt Lake Community College Linda Schmidt, Greenville Technical College Ingrid Scott, Montgomery College A.K Shamma, University