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 us, 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! A00_SULL1772_10_GE_FEP.indd 14/03/16 2:25 PM 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 282 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 383 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 303 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 303 Now Work the ‘Are You Prepared?’ Problems Problems that assess whether you have the Not sure you need the Preparing for This 303, 314 Section review? Work the ‘Are You prerequisite knowledge for the upcoming Prepared?’ problems If you get one wrong, section 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 A00_SULL1772_10_GE_FEP.indd 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 310, 315 336 298, 323 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 214 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 327, 355 320 85, 287, 311 14/03/16 2:25 PM Practice ‘‘Work the Problems’’ Feature 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! 314, 328 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 314 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 314–316 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 316–317 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 317–319 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 319 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 319 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 379–381 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 A01_SULL1772_10_GE_FM.indd 312, 315, 316 29/03/16 4:07 pm 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 378–379 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 379–381 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 382–383 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 383–384 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 A01_SULL1772_10_GE_FM_3-27.indd 377–378 382 383 16/03/16 10:56 AM 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 A01_SULL1772_10_GE_FM_3-27.indd 16/03/16 10:56 AM 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 A01_SULL1772_10_GE_FM_3-27.indd 16/03/16 10:56 AM Precalculus Tenth Edition Global Edition Michael Sullivan Chicago State University A01_SULL1772_10_GE_FM.indd 24/04/17 10:04 AM Editor in Chief: Anne Kelly Acquisitions Editor: Dawn Murrin Assistant Acquisitions Editor, Global Editions: Aditee Agarwal Assistant Editor: Joseph Colella Program Team Lead: Karen Wernholm Program Manager: Chere Bemelmans Project Team Lead: Peter Silvia Project Manager: Peggy McMahon Project Editor, Global Editions: K.K Neelakantan Senior Manufacturing Controller, Global Editions: Trudy Kimber Associate Media Producer: Marielle Guiney Media Production Manager, Global Editions: Vikram Kumar 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, 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 Acknowledgments of third-party content appear on page 1147, 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 M icrosof t and /or its respecti ve suppliers make no represe n tati o ns ab o ut the suitab ility o f the inf ormati on c on tai ned i n the documents a nd related graphics pu blished as part o f the services f or a ny purpo se All such d o cuments a n d related graphics are prov ided “as is ” witho ut warra n ty o f a ny k in d M icro s o f t a n d /or its respective suppliers here by disclaim all warranties and co nditi o ns with regard to this information, including all warranties and conditions of merchantability, whether express, implied or statuto ry, fit ness f o r a particular purpo se, title an d n on - i n frin gemen t In no event shall microso ft and /o r its respective suppliers b e lia b le f o r a ny special, in direct or c o n sequential damages or a ny damages whatsoever resulting from loss of use, data or profits, whether in an action of contract, negligence or other tortious action, arising out of or in connection with the use or performance of information available from the services The documents and related graphics contained herein could include technical inaccuracies or typographical errors Changes are periodically added to the information herein Micro s o f t a nd / o r its respectiv e suppliers may mak e improvements an d /o r cha nges in the pro duct (s ) a nd /o r the pro gram (s ) describ ed herein at a ny time Partial screen sh ots may be v iewed in full within the so ftware versi o n specif ied Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsonglobaleditions.com © Pearson Education Limited 2018 The right of Michael Sullivan to be identified as the author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988 Authorized adaptation from the United States edition, entitled Precalculus, 10th edition, ISBN 978-0-321-97907-0, by Michael Sullivan published by Pearson Education © 2016 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a license permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS All trademarks used herein are the property of their respective owners The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library 10 A01_SULL1772_10_GE_FM.indd Typeset by Cenveo® Publisher Services Printed and bound in Malaysia ISBN-10: 1-292-12177-7 ISBN-13: 978-1-292-12177-2 28/04/17 2:06 PM Contents Three Distinct Series 20 The Contemporary Series 21 Preface to the Instructor 22 Resources for Success 26 Applications Index 28 1 Graphs 35 1.1 The Distance and Midpoint Formulas 36 Use the Distance Formula • Use the Midpoint Formula 1.2 Graphs of Equations in Two Variables; Intercepts; Symmetry 43 Graph Equations by Plotting Points • Find Intercepts from a Graph • Find Intercepts from an Equation • Test an Equation for Symmetry with Respect to the x-Axis, the y-Axis, and the Origin • Know How to Graph Key Equations 1.3 Lines 53 Calculate and Interpret the Slope of a Line • Graph Lines Given a Point and the Slope • Find the Equation of a Vertical Line • Use the Point–Slope Form of a Line; Identify Horizontal Lines • Find the Equation of a Line Given Two Points • Write the Equation of a Line in Slope–Intercept Form • Identify the Slope and y-Intercept of a Line from Its Equation • Graph Lines Written in General Form Using Intercepts • Find Equations of Parallel Lines • Find Equations of Perpendicular Lines 1.4 Circles 68 Write the Standard Form of the Equation of a Circle• Graph a Circle • Work with the General Form of the Equation of a Circle Chapter Review 74 Chapter Test 76 Chapter Project 77 Functions and Their Graphs 78 2.1 Functions 79 Determine Whether a Relation Represents a Function • Find the Value of a Function • Find the Difference Quotient of a Function • Find the Domain of a Function Defined by an Equation • Form the Sum, Difference, Product, and Quotient of Two Functions 2.2 The Graph of a Function 94 Identify the Graph of a Function • Obtain Information from or about the Graph of a Function 2.3 Properties of Functions 103 Determine Even and Odd Functions from a Graph • Identify Even and Odd Functions from an Equation • Use a Graph to Determine Where a Function Is Increasing, Decreasing, or Constant • Use a Graph to 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 a Function Is Increasing or Decreasing • Find the Average Rate of Change of a Function A01_SULL1772_10_GE_FM_3-27.indd 16/03/16 10:56 AM Answers Section 4.1 1063 (b) 12, - 82 (c) x = (d) x-intercepts: (e) Maximum value; 21 5x͉ x … or x Ú 6; ( - q , 4 ∪ 6, q ) (a) C(m) = 0.15m + 129.50 (b) $258.50 (c) 562 miles - 26 + 26 , ; y-intercept: 3 y 10 (0, 4) (4, 4) 10 x (3.63, 0) (2, ؊8) x؍2 (0.37, 0) Cumulative Review (page 199) 1 22; a , b - 2, - 12 and (2, 3) are on the graph 2 3 e x ` x Ú - f ; c - , q b 5 ؊ 13 y = - x + 2 y = - 2x + (؊1, 4) y 1x - 22 + 1y + 42 = 25 y 10 y (2, 1) (3, 5) x (2, ؊2) (؊3, ؊4) x 10 x (7, ؊4) (2, ؊4) (2, ؊9) f 10 Yes 11 (a) No (b) - 1; ( - 2, - 1) is on the graph. (c) - 8; ( - 8, 2) is on the graph. 12 Neither 13 Local maximum value is 5.30 and occurs at x = - 1.29 Local minimum value is - 3.30 and occurs at x = 1.29 Increasing: - 4, - 1.292 and (1.29, 4); Decreasing: - 1.29, 1.292 Yes (a) - 3 (b) x2 - 4x - 2 (c) x2 + 4x + 1 (d) - x2 + 4x - 1 (e) x2 - 3 (f) 2x + h - 4 e z ` z ≠ 14 (a) - 4 (b) 5x ͉ x - or - 4, q 15 (a) Domain: 5x ͉ - … x … 6; Range: 5y ͉ - … y … (b) - 1, 02, 10, - 12, 11, 02 (c) y-axis (d) 1 (e) - and 4 (f) 5x ͉ - x 6 (g) y (؊4, 5) (4, 5) (2, 3) (؊2, 3) (1, 2) (؊1, 2) x (0, 1) (h) y (؊4, 3) (؊2, 1) (؊1, 0) (0, ؊1) (i) y 10 (؊4, 6) (4, 3) (2, 1) x (1, 0) (؊2, 2) (؊1, 0) (0, ؊2) (j) Even (k) (0, 4) (4, 6) (2, 2) x (1, 0) Chapter 4 Polynomial and Rational Functions 4.1 Assess Your Understanding (page 218) smooth; continuous touches - 1, 12; 10, 02; 11, 12 10 r is a real zero of f ; r is an x-intercept of the graph of f ; x - r is a factor of f 11 turning points 12 y = 3x4 13 q ; - q 14 As x increases in the positive direction, f 1x2 decreases without bound. 15 b 16 d 1 17 Yes; degree 3; f 1x2 = x3 + 4x; leading term: x3; constant term: 0 19 Yes; degree 1; h 1x2 = - x; leading term: - x; constant term: 2 21 No; x is raised to the –1 power 23 No; x is raised to the power 25 No; x is raised to the - 1 and - powers 27 Yes; degree 5; G1x2 = - 3x5 - 18x4 - 36x3 - 24x2; leading term: - 3x5 29 y 31 (0, 1) (؊2, 1) x (؊1, 0) y (؊1, 3) ؊5 33 (1, 3) (0, 2) x ؊1, 41 y ؊5 (0, ؊2.5) x (2, ؊1.5) ؊5 (1, ؊2) Z03_SULL1772_10_GE_APPB_ANS.indd 1063 1, x ؊5 (0, 0) ؊5 39 35 y 37 y y (؊2, 3) (؊1, 2) (؊3, 2) x ؊5 ؊5 (؊3, ؊2) ؊5 (؊1, 1) (0, 0) x (1, ؊1) (؊2, ؊3) ؊5 ؊5 y x (؊1, ؊2) 43 f 1x2 = x - 3x - x + for a = 1 45 f 1x2 = x + 2x - 8x for a = 47 f 1x2 = x4 - 3x3 - 15x2 + 19x + 30 for a = 1 49 f 1x2 = x3 - 12x - 16 for a = 1 16 32 496 512 960 x + x x 51 f(x) = - x3 + x 53 f(x) = - x4 3 49 49 49 49 49 55 f 1x2 = - 3x5 + 12x4 + 6x3 - 36x2 - 27x 57 (a) 7, multiplicity 1; - 3, multiplicity 2 (b) Graph touches the x-axis at - and crosses it at 7. (c) 2 (d) y = 3x3 02/05/17 2:08 PM 1064 Answers Section 4.1 59 (a) 3, multiplicity 1 (b) Graph crosses the x-axis at 3. (c) 6 (d) y = 2x7 61 (a) , multiplicity 2; 1, multiplicity 3 (b) Graph touches the and crosses the x-axis at 1. (c) 4 (d) y = x5 63 (a) - 23, multiplicity 2; 2, multiplicity 4 (b) Graph touches the x-axis at - 23 and x-axis at at 2. (c) 5 (d) y = x6 65 (a) No real zeros. (b) Graph neither touches nor crosses the x-axis. (c) 5 (d) y = - 2x6 67 (a) - 23, 23, 0, multiplicity 1 (b) Graph crosses the x-axis at - 13, 13, 0. (c) 2 (d) y = 4x3 69 Could be; zeros: - 1, 1, 2; Least degree is 3. 71 Cannot be the graph of a polynomial; gap at x = 0 73 f 1x2 = x 1x - 12 1x - 22 75 f 1x2 = - 1x + 12 1x - 12 1x - 22 77 f 1x2 = 0.2 1x + 42 1x + 12 1x - 32 79 f 1x2 = x2 1x + 32 1x + 12 1x - 22 81 Step 1: y = x3 83 Step 1: y = x3 Step 2: x-intercepts: 0, 3; y-intercept: Step 2: x@intercepts: 1, - 3; y@intercepts: - Step 3: 0: multiplicity 2, touches; 3: multiplicity 1, crosses Step 3: - 3: multiplicity 2, touches; 1: multiplicity 1, crosses Step 4: At most turning points Step 4: At most turning points Step 5: f 1- 12 = - 4; f 122 = - 4; f 142 = 16 Step 5: f - 42 = - 5; f - 12 = - 8; f 122 = 25 y 24 y 30 (4, 16) (3, 0) x (2, ؊4) (0, 0) (؊1, ؊4) 85 Step 1: y = - x4 Step 2: x-intercepts: - 4, 1; y-intercept: Step 3: - 4: multiplicity 1, crosses; 1: multiplicity 3, crosses Step 4: 3 Step 5: f 1- 52 = - 108; f - 32 = 32; f 132 = - 28 y (؊3, 32) 40 (؊4, 0) (2, 25) (؊3, 0) (1, 0) x (؊4, ؊5) (0, ؊9) (؊1, ؊8) (0, 2) 87 Step 1: y = x3 Step 2: x-intercepts: 1, - 4, 3; y-intercept: - 12 Step 3: - 4, 1, 3: multiplicity 1, crosses Step 4: 2 Step 5: f 1- 52 = - 48; f - 22 = 30; f 122 = - 6; f 142 = 24 y (0, 12) (؊2, 30) 40 (4, 24) (3, 0) x (1, 0) (2, ؊6) (؊5, ؊48) (؊4, 0) (1, 0) x (3, ؊28) (؊5, ؊108) 89 Step 1: y = x4 Step 2: x-intercepts: 0, 3, - 4; y-intercepts: Step 3: - 4, 3: multiplicity 1, crosses; 0: multiplicity 2, touches Step 4: 3 y Step 5: f 1- 32 = - 54; f - 22 = - 40; 20 (3, 0) (؊4, 0) f 112 = - 10; f 122 = - 24 (0, 0) (؊3, ؊54) x (2, ؊24) 91 Step 1: y = x4 Step 2: x-intercepts: 4, - 2; y-intercept: 64 Step 3: - 2, 4: multiplicity 2, touches Step 4: 3 Step 5: f 1- 32 = 49; f 122 = 64; f 152 = 49 93 Step 1: y = x Step 2: x-intercepts: 0, 3, 1; y-intercept: Step 3: 0: multiplicity 2, touches; 1, 3: multiplicity 1, crosses Step 4: 3 Step 5: f - 12 = 8; f a b = ; f 122 = - 4; f 142 = 48 16 y , 16 (4, 48) (1, 0) (3, 0) x y (2, 64) 90 (5, 49) (4, 0) (؊2, ؊40) (؊1, 8) (0, 0) (؊3, 49) (1, ؊10) 54 (0, 64) (؊2, 0) x 95 Step 1: y = x Step 2: x-intercepts: 2, - 2, - 4; y-intercept: 32 Step 3: 2: multiplicity 2, touches; - 2, - 4: multiplicity 1, crosses Step 4: 3 Step 5: f - 52 = 147; f - 32 = - 25; f - 12 = 27; f 132 = 35 y 200 (؊2, 0) (؊4, 0) (؊3, ؊25) (0, 32) (3, 35) (2, 0) x (2, ؊4) 97 Step 1: y = x5 Step 2: x-intercepts: 0, - 4; y-intercept: Step 3: 0: multiplicity 2, touches; - 4: multiplicity 1, crosses Step 4: 4 Step 5: f - 52 = - 650; f - 32 = 90; f - 22 = 40; f 112 = 10 y 160 (؊3, 90) (؊4, 0) (؊2, 40) (1, 10) (0, 0) x Z03_SULL1772_10_GE_APPB_ANS.indd 1064 02/05/17 2:08 PM Answers Section 4.1 1065 101 Step 1: y = x3 Step 2: 99 Step 1: y = x3 Step 2: 103 Step 1: y = x4 Step 2: 60 15 Ϫ3 Ϫ2 Ϫ2 Ϫ40 Ϫ30 Step 3: x-intercepts: - 1.26, - 0.20, 1.26 Step 3: x-intercepts: - 0.9, 4.71 y-intercept: - 3.8151 y-intercept: - 0.31752 Step 4: Step 4: Step 5: - 0.80, 0.572; 10.66, - 0.992 Step 6: y (؊0.5, 0.40) 2.5 (؊0.20, 0) (؊0.80, 0.57) (1.5, 1.13) 2.5 x (1.26, 0) (0.66, ؊0.99) (؊1.26, 0) (؊1.5, ؊0.86) (0, ؊0.32) Step 7: Domain: - q , q 2; Range: - q , q Step 8: Increasing on - q , - 0.802 and 10.66, q Decreasing on - 0.80, 0.662 105 Step 1: y = - 1.2x4 Step 2: Ϫ5 Ϫ3 Ϫ5 Step 3: x-intercepts: - 1.47, 0.91; y-intercept: Step 4: Step 5: 10.81, 3.212 Step 6: y (؊0.81, 3.21) (؊1.47, 0) ؊3 (؊1.7, ؊3.63) (0, 2) (0.91, 0) 3x (1.25, ؊2.31) ؊5 Step 7: Domain: - q , q 2; Range: - q , 3.21 Step 8: Increasing on - q , - 0.812 Decreasing on - 0.81, q Z03_SULL1772_10_GE_APPB_ANS.indd 1065 Step 5: - 0.9, 02; 12.84, - 26.162 Step 6: y (0, ؊3.82) 15 (؊0.9, 0) ؊4 (؊2, ؊8.12) ؊30 (5, 10.10) x (2, ؊22.79) (2.84, ؊26.16) Step 7: Domain: - q , q 2; Range: - q , q Step 8: Increasing on - q , - 0.92 and 12.84, q Decreasing on - 0.9, 2.842 107 f 1x2 = - x 1x + 22 1x - 22 Step 1: y = - x3 Step 2: x-intercepts: - 2, 0, 2; y-intercept: Step 3: - 2, 0, 2: multiplicity 1, crosses Step 4: At most turning points Step 5: f - 32 = 15; f - 12 = - 3; f 112 = 3; f 132 = - 15 y 20 (0, 0) (؊3, 15) (1, 3) (؊2, 0) (2, 0) x (؊1, ؊3) (3, ؊15) 111 f 1x2 = 2x 1x + 62 1x - 22 1x + 22 Step 1: y = 2x4 Step 2: x-intercepts: - 6, - 2, 0, 2; y-intercept: Step 3: - 6, - 2, 0, 2: multiplicity 1, crosses Step 4: At most turning points Step 5: f - 72 = 630; f - 42 = - 192; f - 12 = 30; f 112 = - 42; f 132 = 270 y 700 (؊7, 630) (؊1, 30) (؊6, 0) (؊2, 0) (؊4, ؊192) (0, 0) (3, 270) (2, 0) x (1, ؊42) Step 3: x-intercepts: - 3.90, - 1.82, 1.82, 3.90 y-intercept: 50.2619 Step 4: Step 5: 10, 50.262; - 3.04, - 35.302, 13.04, - 35.302 Step 6: y 60 (؊1.82, 0) (؊4, 10.26) (؊3.90, 0) ؊5 (؊2.5, ؊26.3) ؊40 (؊3.04, ؊35.30) (0, 50.26) (4.25, 42.36) (1.82, 0) (3.90, 0) x (2.5, ؊26.3) (3.04, ؊35.30) Step 7: Domain: - q , q 2; Range: - 35.30, q Step 8: Increasing on 1- 3.04, 02 and 13.04, q Decreasing on - q , - 3.042 and 10, 3.042 109 f 1x2 = x 1x + 42 1x - 32 Step 1: y = x3 Step 2: x-intercepts: - 4, 0, 3; y-intercept: Step 3: - 4, 0, 3: multiplicity 1, crosses Step 4: At most turning points Step 5: f - 52 = - 40; f - 22 = 20; f 122 = - 12; f 142 = 32 y 50 (؊2, 20) (؊4, 0) (؊5, ؊40) (4, 32) (3, 0) x (0, 0) (2, ؊12) 113 f 1x2 = - x2 1x + 12 1x - 12 Step 1: y = - x5 Step 2: x-intercepts: - 1, 0, 1; y-intercept: Step 3: 1: multiplicity 1, crosses; - 1, 0: multiplicity 2, touches Step 4: At most turning points Step 5: f 1-1.52 = 1.40625; f 1- 0.542 = 0.10; f 10.742 = 0.43; f 11.22 = - 1.39392 y 2.5 (؊1.5, 1.40625) (؊1, 0) (؊0.54, 0.10) (0, 0) (0.74, 0.43) (1, 0) 2.5 x (1.2, ؊1.39392) 02/05/17 2:08 PM 1066 Answers Section 4.1 115 f 1x2 = 1x + 32 1x - 12 1x - 42 117 f 1x2 = - 1x + 52 1x - 22 1x - 42 119 (a) - 3, 2 (b) - 6, - 123 (a) T 121 (a) H 60 54 48 42 36 30 50 40 30 20 10 10 x 12 18 24 x The relation appears to be cubic (b) 2.4°/h (c) 1°/h (d) T 1x2 = - 0.01308x3 + 0.4674x2 - 3.4159x + 41.1929; 53.9°F (e) The relation appears to be cubic (b) H 1x2 = 0.3948x3 - 5.9563x2 + 26.1965x - 7.4127 (c) ≈ 24 (d) 60 35 0 25 10 27 (f) The predicted temperature at midnight is 41.2°F (e) ≈ 54; no 125 (a) (b) (c) (d) As more terms are added, the values of the polynomial function get closer to the values of f The approximations near are better than those near - or 131 1a2 - 1d2 134 y = - - - 27 - + 27 11 135 5x͉ x ≠ - 136 137 ( - 7, 3) x , 5 2 4.2 Assess Your Understanding (page 230) F horizontal asymptote vertical asymptote proper T 10 F 11 y = 0 12 T 13 d 14 a 15 All real numbers except - 3; 1 x x ≠ - 17 All real numbers except and - 4; x x ≠ 2, x ≠ - 19 All real numbers except - and ; e x ` x ≠ - 2, x ≠ f 3 21 All real numbers except and - 1; x x ≠ - 1, x ≠ 23 All real numbers 25 All real numbers except - x x ≠ - 27 (a) Domain: 29 (a) Domain: 31 (a) Domain: 33 (a) y 10 5x x 5x x 5x x ≠ ; range: y y ≠ (b) 10, 02 (c) y = 1 (d) x = 2 (e) None ≠ ; Range: y y … - or y Ú (b) None (c) None (d) x = 0 (e) y = - x ≠ - 1, x ≠ Range: All real numbers (b) 10, 02 (c) y = 0 (d) x = - 1, x = 1 (e) None 35 (a) 37 (a) y 10 (؊3, 2) y y؍0 (؊1, 4) (1, 4) ؊5 y؍3 x؍0 (0, 1) x ؊5 (b) Domain: 5x͉ x ≠ ; Range: 5y͉ y (c) Vertical asymptote: x = 0; Horizontal asymptote: y = 39 (a) y؍1 y x؍1 (2, 2) ؊5 (0, 0) x ؊5 (b) Domain: 5x͉ x ≠ ; Range: 5y͉ y ≠ (c) Vertical asymptote: x = 1; Horizontal asymptote: y = Z03_SULL1772_10_GE_APPB_ANS.indd 1066 y؍0 (2, 1) x؍1 x (b) Domain: 5x͉ x ≠ ; range: 5y͉ y (c) Vertical asymptote: x = 1; horizontal asymptote: y = y 41 (a) (؊2, 3) ؊5 x ؍؊1 y؍2 (0, 1) x ؊5 (b) Domain: 5x͉ x ≠ - ; Range: 5y͉ y ≠ (c) Vertical asymptote: x = - 1; Horizontal asymptote: y = ؊2 (؊1, 2) x x ؍؊2 (b) Domain: 5x͉ x ≠ - ; Range: 5y͉ y (c) Vertical asymptote: x = - 2; Horizontal asymptote: y = y 43 (a) (0, 2) (−4, 0) −5 (0, −2) −5 (5, 3) x (5, −3) (b) Domain: 5x͉ x ≠ ; Range: 5y͉ y ≠ (c) Vertical asymptote: x = 0; Horizontal asymptote: y = 05/05/17 11:58 AM Answers Section 4.3 1067 45 Vertical asymptote: x = - 4; horizontal asymptote: y = 3 47 Vertical asymptote: x = 3; oblique asymptote: y = x + 5 49 Vertical asymptotes: x = 1, x = - 1; horizontal asymptote: y = 0 51 Vertical asymptote: x = - ; Horizontal asymptote: y = 2 53 Vertical asymptote: none; oblique asymptote: y = 2x + 7 55 Vertical asymptote: x = 0; No horizontal or oblique asymptote 59 (a) 10 69 x@axis symmetry 70 - 3, 112, 12, - 42 y x؍1 10 (2, 7) 67 x = 5 68 - 30 61 (a) R1x2 = + = 5a b + x - x - (b) Rtot y؍2 ؊5 (0, ؊3) 10 15 20 25 R2 (b) Horizontal: Rtot = 10; as the resistance of R2 increases without bound, the total resistance approaches 10 ohms, the resistance R1 (c) R1 ≈ 103.5 ohms y؍2 x ؊10 (c) Vertical asymptote: x = 1; horizontal asymptote: y = 4.3 Assess Your Understanding (page 245) False c a (a) 5x͉ x ≠ (b) 0 True Domain: {x ͉ x ≠ 0, x ≠ - 4} R is in lowest terms 3 no y-intercept; x-intercept: - R is in lowest terms; vertical asymptotes: x = 0, x = - 4 5 Horizontal asymptote: y = 0, intersected at - 1, 02 ؊4 Interval (؊ؕ, ؊4) (؊4, ؊1) Number Chosen ؊5 ؊2 Value of R R(؊5) ؍؊ 45 Point on Graph ( ) ( )؍ ؊2 ؊7 R(1) ؍ Below x-axis Above x-axis ( ( ( ) ) ) ؊2,؊7 ؊5, ؊ 1, 1, ؊2, Above x-axis ؊2, 4 ؊5, ؊ R x؍0 x ؍؊4 y 2.5 (0, ؕ) ؊2 R(؊2) ؍ Location of Graph Below x-axis (؊1, 0) ؊1 x 5 y؍0 ؊ ,؊ (؊1, 0) 1x + 22 ; Domain: 5x ͉ x ≠ 2 R is in lowest terms 3 y-intercept: - x-intercept: - x - R is in lowest terms; vertical asymptote: x = 1 Horizontal asymptote: y = 2, not intersected R1x2 = 7 ؊2 Interval (؊ؕ, ؊2) (؊2, 1) Number Chosen ؊3 Value of R R(؊3) ؍2 R(0) ؍؊4 R(2) ؍8 Below x-axis Above x-axis (0, ؊4) (2, 8) Location of Graph Above x-axis (؊3, ) Point on Graph y x؍1 (2, 8) 10 (4, 4) y؍2 x (؊2, 0) (0, ؊4) ؊10 (3, 5) (1, ؕ) ; Domain: 5x ͉ x ≠ - 2, x ≠ 2 R is in lowest terms 3 y-intercept: - ; no x-intercept 1x - 32 1x + 22 4 R is in lowest terms; vertical asymptotes: x = - 2, x = 3 5 Horizontal asymptote: y = 0, not intersected 6 7 x ؍؊2 y ؊2 11 Interval (؊ؕ, ؊2) (؊2, 3) (3, ؕ) Number Chosen ؊3 Value of R R(؊3) ؍1 R(0) ؍؊1 R(4) ؍1 Location of Graph Above x-axis Below x-axis Above x-axis Point on Graph (0, ؊1) (4, 1) (؊3, 1) x؍3 (؊3, 1) ؊5 (0, ؊1) ؊2 (4, 1) y؍0 x (1, ؊1) 1x + x + 12 1x - x + 12 ; Domain: 5x͉ x ≠ - 1, x ≠ 2 P is in lowest terms 3 y-intercept: - 1; no x-intercept 1x + 12 1x - 12 4 P is in lowest terms; vertical asymptotes: x = - 1, x = 1 5 No horizontal or oblique asymptote y x؍1 ؊1 7 6 13 P1x2 = Interval (؊ؕ, ؊1) (؊1, 1) (1, ؕ) Number Chosen ؊2 Value of P P(؊2) ؍7 P(0) ؍؊1 P(2) ؍7 Location of Graph Above x-axis Below x-axis Above x-axis Point on Graph (0, ؊1) (2, 7) (؊2, 7) Z03_SULL1772_10_GE_APPB_ANS.indd 1067 (؊2, 7) (2, 7) (0, ؊1) x x ؍؊1 02/05/17 2:08 PM 1068 Answers Section 4.3 1x - 12 1x2 + x + 12 15 H1x2 = 1x + 32 1x - 32 ; x-intercept: ; Domain: 5x ͉ x ≠ - 3, x ≠ 2 H is in lowest terms 3 y-intercept: 1 4 H is in lowest terms; vertical asymptotes: x = 3, x = - 3 5 Oblique asymptote: y = x, intersected at a , b 9 6 7 y ؊3 Interval (؊ؕ, ؊3) (؊3, 1) (1, 3) (3, ؕ) Number Chosen ؊4 Value of H H(؊4) Ϸ ؊9.3 H(0) ؍9 H(2) ؍؊1.4 H(4) ؍9 Above x-axis Below x-axis Above x-axis (0, ) (2, ؊1.4) (4, 9) Location of Graph Below x-axis Point on Graph (؊4, ؊9.3) 1 0, (4, 9) y؍x (1, 0) 10 x (2, ؊1.4) (؊4, ؊9.3) x ؍؊3 x ؍3 1x + 42 1x - 32 ; Domain: 5x x ≠ - 2, x ≠ 2 R is in lowest terms 3 y-intercept: 3; x-intercepts: - 4, 1x + 22 1x - 22 4 R is in lowest terms; vertical asymptotes: x = - 2, x = 2 5 Horizontal asymptote: y = 1, intersected at 18, 12 ؊4 ؊2 y 6 7 17 R1x2 = Interval (؊ؕ, ؊4) (؊4, ؊2) (؊2, 2) (2, 3) (3, ؕ) Number Chosen ؊7 ؊3 2.5 Value of R R(؊7) ؍23 R(؊3) ؍؊1.2 R(0) ؍3 R(2.5) ؍؊1.44 R(8) ؍1 Below x-axis Above x-axis Below x-axis Above x-axis (؊3, ؊1.2) (0, 3) (2.5, ؊1.44) (8, 1) Location of Graph Above x-axis Point on Graph (؊7, ) (0, 3) (؊4, 0) x؍2 (8, 1) 10 x (3, 0) ؊5 y؍1 ؊5 x ؍؊2 3x ; Domain: 5x ͉ x ≠ - 1, x ≠ 2 G is in lowest terms 3 y-intercept: 0; x-intercept: 1x + 12 1x - 12 4 G is in lowest terms; vertical asymptotes: x = - 1, x = 1 5 Horizontal asymptote: y = 0, intersected at 10, 02 y 7 ؊1 6 19 G1x2 = Interval (؊ؕ, ؊1) (؊1, 0) Number Chosen ؊2 ؊2 Value of G G(؊2) ؍؊2 G ؊2 Location of Graph Below x-axis Point on Graph (؊2, ؊2) ( )؍2 (0, 1) (1, ؕ) 2 ( ) ؍؊2 G G(2) ؍2 Above x-axis Below x-axis Above x-axis (؊ , 2) ( , ؊2) (2, 2) 2 (2, 2) x ؍؊1 (0, 0) x x؍1 (؊2, ؊2) ؊5 y؍0 -4 ; Domain: 5x ͉ x ≠ - 3, x ≠ - 1, x ≠ 2 R is in lowest terms 3 y-intercept: ; x-intercepts: None 1x + 12 1x + 32 1x - 32 4 R is in lowest terms; vertical asymptotes: x = - 3, x = - 1, x = 3 5 Horizontal asymptote: y = 0, not intersected ؊3 ؊1 7 6 21 R1x2 = 0, Interval (؊ؕ, ؊3) (؊3, ؊1) (؊1, 3) (3 ؕ) Number Chosen ؊4 ؊2 Value of R R(؊4) Ϸ 0.19 R(؊2) ؍؊0.8 R(0) ؍ R(4) Ϸ ؊0.11 Location of Graph Above x-axis Below x-axis Above x-axis Below x-axis Point on Graph (؊2, ؊0.8) (0, ) (4, ؊0.11) (؊4, 0.19) 9 y x؍3 (؊4, 0.19) ؊5 (؊2, ؊0.8) (1, 0.25) y؍0 x ؊2 (4, ؊0.11) x ؍؊3 x ؍؊1 x2 + ; Domain: 5x ͉ x ≠ - 1, x ≠ 2 H is in lowest terms 3 y-intercept: - ; no x-intercepts 1x + 12 1x + 12 1x - 12 4 H is in lowest terms; vertical asymptotes: x = - and x = 1 5 Horizontal asymptote y = 0, not intersected 7 6 y ؊1 23 H1x2 = Interval Number Chosen Value of H Location of Graph Point on Graph (؊ؕ, ؊1) ؊2 H(؊2) ؍15 Above x-axis (؊2, 15 ) (؊1, 1) H(0) ؍؊4 Below x-axis (0, ؊4) ؊2, (1, ؕ) H(2) ؍15 Above x-axis (2, 15 ) 15 15 2, x ؊5 y؍0 (0, ؊4) x ؍؊1 x ؍1 1x + 22 1x + 12 ; Domain: 5x ͉ x ≠ - 2 F is in lowest terms. 3 y-intercept: - 2; x-intercept: - 2, - x - 4 F is in lowest terms; vertical asymptote: x = 1 5 Oblique asymptote: y = x + 4; not intersected y 6 7 ؊2 ؊1 25 F1x2 = Interval (؊ؕ, ؊2) (؊2, ؊1) (؊1, 1) (1, ؕ) Number Chosen ؊3 ؊1.5 Value of F F(؊3) ؍؊0.5 F(؊1.5) ؍0.1 F(0) ؍؊2 F(2) ؍12 Location of Graph Below x-axis Above x-axis Below x-axis Above x-axis Point on Graph (؊1.5, 0.1) (0, ؊2) (2, 12) (؊3, ؊0.5) Z03_SULL1772_10_GE_APPB_ANS.indd 1068 16 (21, 0) (2, 12) y5x14 (3, 10) (22, 0) x (0, 22) x51 02/05/17 2:08 PM Answers Section 4.3 1069 1x - 42 1x + 32 12 ; Domain: 5x͉ x ≠ - 2 R is in lowest terms. 3 y-intercept: - ; x-intercept: - and x + 5 4 R is in lowest terms; vertical asymptote: x = - 5 5 Oblique asymptote: y = x - 6; not intersected 6 7 y ؊5 ؊3 27 R1x2 = Interval (؊ؕ, ؊5) (؊5, ؊3) (؊3, 4) (4, ؕ) Number Chosen ؊7 ؊4 Value of R R(؊7) ؍؊22 R(؊4) ؍8 R(0) ؍؊2.4 R(5) ؍0.8 Location of Graph Below x-axis Above x-axis Below x-axis Above x-axis Point on Graph (؊4, 8) (0, ؊2.4) (5, 0.8) (؊7, ؊22) 10 (؊3, 0) y؍x؊6 10 x (4, 0) (0, ؊2.4) ؊10 (؊7, ؊22) ؊40 x ؍؊5 1x + 32 1x - 42 ; Domain: 5x͉ x ≠ - 2 G is in lowest terms. 3 y-intercept: - 12; x-intercept: - and x + 4 G is in lowest terms; vertical asymptote: x = - 1 5 Oblique asymptote: y = x - 2; not intersected y 6 7 ؊3 ؊1 29 G1x2 = Interval (؊ؕ, ؊3) Number Chosen ؊4 Value of G G(؊4) ؍؊ –3 Location of Graph Below x-axis Point on Graph (؊4, ؊ ) 8– (؊3, ؊1) (؊1, 4) (4, ؕ) ؊2 (؊2, 6) (؊3, 0) ؊5 y؍x؊2 G(؊2) ؍6 G(0) ؍؊12 G(5) – ؍3 Above x-axis Below x-axis Above x-axis (؊2, 6) (0, ؊12) (4, 0) x (0, ؊12) x ؍؊1 (5, ) 4– 31 Domain: 5x͉ x ≠ - R is in lowest terms y-intercept: 0; x-intercepts: 0, 1 4 Vertical asymptote: x = - 5 Horizontal asymptote: y = 1, not intersected 6 See enlarged ؊3 Interval (؊ؕ, ؊3) (؊3, 0) (0, 1) (1, ؕ) Number Chosen ؊4 ؊1 1– 2 Value of R R(؊4) ؍100 R(؊1) ؍؊0.5 R Location of Graph Above x-axis Below x-axis Above x-axis Point on Graph (؊1, ؊0.5) (؊4, 100) 1x + 42 1x - 32 33 R1x2 = 1x + 22 1x - 32 ( ( ) Ϸ 0.003 1– 1– 2, ) 0.003 y R (2) ؍0.016 (؊ؕ, ؊4) Number Chosen ؊5 Value of R R(؊5) ؍ Location of Graph Above x-axis Point on Graph (؊5, ) 1– (2, 0.016) (؊2, 3) (3, ؕ) ؊3 R(؊3) ؍؊1 R(0) ؍2 R(4) – ؍3 Below x-axis Above x-axis Above x-axis (0, 2) (0, 0) x ؍؊3 0.01 (1, 0) 1.25 x Enlarged view x + 3 y-intercept: 2; x-intercept: - x + y ؊5, (؊4, ؊2) (؊3, ؊1) y؍1 10 x (1, 0) Above x-axis ; Domain: 5x͉ x ≠ - 2, x ≠ 2 In lowest terms, R1x2 = 1– y (0, 0) 4 Vertical asymptote: x = - 2; hole at a3, b 5 Horizontal asymptote: y = 1, not intersected ؊2 ؊4 6 7 Interval view at right 10 (؊4, 0) 3, (0, 2) y؍1 10 x (؊3, ؊1) 4, x ؍؊2 (4, ) 4– 14x + 32 12x + 52 4x + 3 ; Domain: e x ` x ≠ - , x ≠ f 2 In lowest terms, R1x2 = 3 y-intercept: - ; x-intercept: x - 14 4 Vertical asymptote: x = 3; hole at a - , b 5 Horizontal asymptote: y = 4; not intersected 11 y x؍3 ؊4 ؊2.5 6 7 20 35 R1x2 = 12x + 52 1x - 32 Interval (؊ؕ, ؊2.5) (؊2.5 , ؊ ) (؊ , 3) (3, ؕ) Number Chosen ؊3 ؊1 R(0) ؍؊1 R(6) ؍9 Value of R R(؊3) ؍ Location of Graph Above x-axis Point on Graph (؊3, ) 3 R(؊1) ؍ 4 Above x-axis Below x-axis Above x-axis (؊1, ) (0, ؊1) (6, 9) 14 ؊ , 11 (6, 9) y؍4 (0, ؊1) 12 x ؊4 ؊ ,0 1x + 62 1x - 52 ; Domain: 5x͉ x ≠ - 6 2 In lowest terms, R1x2 = x - 5 3 y-intercept: - 5; x-intercept: x + 4 Vertical asymptote: none; hole at - 6, - 112 5 Oblique asymptote: y = x - intersected at all points except x = - y ؊6 6 7 37 R1x2 = (؊ؕ, ؊6) (؊6, 5) (5, ؕ) Number Chosen ؊7 Value of R R(؊7) ؍؊12 R(0) ؍؊5 R(6) ؍1 Location of Graph Below x-axis Below x-axis Above x-axis Point on Graph (0, ؊5) (6, 1) Interval (؊7, ؊12) Z03_SULL1772_10_GE_APPB_ANS.indd 1069 10 x ؊10 (؊6, ؊11) (5, 0) (0, ؊5) ؊12 02/05/17 2:08 PM 1070 Answers Section 4.3 - 1x - 12 ; Domain: 5x͉ x ≠ - 1, x ≠ 2 In lowest terms, H1x2 = 3 y-intercept: - ; no x-intercept 1x + 12 1x - 12 x + 4 Vertical asymptote: x = - 1; hole at - 1, 12 5 Horizontal asymptote: y = 0; not intersected 39 H1x2 = 6 7 Interval (1, 4) (؊ؕ, 1) 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) y x=1 Q5, 14R (0, 4) (4, ؕ) y=1 −5 x (4, 0) (2, −2) −5 1x - 52 1x + 32 x - 5 ; Domain: {x͉ x ≠ - 3} 2 In lowest terms, F1x2 = 3 y-intercept: - ; x-intercept: 1x + 32 1x + 32 x + 3 4 Vertical asymptote: x = - 3; 5 Horizontal asymptote: y = 1; not intersected 23 6 7 x = −3 y 41 F1x2 = Interval (2`, 23) (23, 5) (5, `) Number Chosen 24 Value of H 23 Location of Graph Above x-axis Below x-axis Above x-axis (24, 9) 0, Point on Graph ( ) (−4, 9) 10 Q6, 19R y=1 −10 10 x (5, 0) Q0, − 53R (6 , ) - x ; Domain: 5x͉ x ≠ 2 G is in lowest terms 3 y-intercept: 2; x-intercept: 1x - 12 4 Vertical asymptote: x = 1 5 Horizontal asymptote: y = 0; intersected at (2, 0) 6 7 43 G1x2 = Interval (؊ؕ, 1) (1, 2) (2, ؕ) 3 Number Chosen y x=1 (0, 2) y=0 −3 Value of H 2 24 Location of Graph Above x-axis Above x-axis Below x-axis Point on Graph (0, 2) ( , 2) (3 , ) Q32 , 2R −1 (2, 0) x Q3, − 14R 2x2 + ; Domain: 5x͉ x ≠ 2 f is in lowest terms 3 No y-intercept; no x-intercepts x 4 Vertical asymptote: x = 0 5 Oblique asymptote: y = 2x; not intersected 6 7 45 f 1x2 = Interval (؊ؕ, 0) (0, ؕ) Number Chosen ؊1 Value of f f (؊1) ؍؊11 f (1) ؍11 Point on Graph (؊1, ؊11) y ؍2x x (؊1, ؊11) ؊15 x؍0 (؊3, ؊9) Above x-axis Location of Graph Below x-axis y 15 (1, 11) (3, 9) ؊5 (1, 11) 1x + 22 1x2 - 2x + 42 2x3 + 16 = ; Domain: 5x͉ x ≠ 2 f is in lowest terms 3 No y-intercept; x-intercept: - x x 4 Vertical asymptote: x = 0 5 No horizontal or oblique asymptote y 22 6 7 47 f 1x2 = Interval (2`, 22) (22, 0) (0, `) Number Chosen 23 21 Value of f f (23) 16 f (21) 214 f (1) 18 Below x-axis Above x-axis Location of Graph Above x-axis Point on Graph (23, 16) (21, 214) 18 (−2, 0) −6 x x؍0 (1, 18) 2x4 + ; Domain: 5x͉ x ≠ 2 f is in lowest terms 3 no y-intercept; no x-intercepts x3 4 Vertical asymptote: x = 0 5 Oblique asymptote: y = 2x; not intersected 6 49 f 1x2 = (2, 16) (1, 18) Interval (؊ؕ, 0) (0, ؕ) Number Chosen ؊2 Value of f f (؊2) ؍؊5.125 Point on Graph (؊2, ؊5.125) 51 One possibility: R1x2 = x f(2) ؍5.125 y ؍2x (2, 5.125) 10 x (؊2, ؊5.125) ؊10 x؍0 (2, 5.125) x - Z03_SULL1772_10_GE_APPB_ANS.indd 1070 ؊10 Above x-axis Location of Graph Below x-axis y 10 53 One possibility: R1x2 = 1x + 22 1x - 12 1x + 32 1x - 42 02/05/17 2:08 PM Answers Historical Problems 1071 55 (a) t-axis; C1t2 S (b) 57 (a) C1x2 = 16x + (b) x (c) 0.4 5000 + 100 x 12 0 (c) 0.71 h after injection 61 (a) C1r2 = 12pr + (b) 300 0 60 (c) 2784.95 in.2 (d) Approximately 17.7 ft by 56.6 ft (d) 21.54 in * 21.54 in * 21.54 in (longer side parallel to river) (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 (2, - 5) 71 y = x - 4 72 73 g(3) = 4000 r 6000 0 (b) 40,000 x 10,000 10,000 0 59 (a) S 1x2 = 2x2 + 10 The cost is smallest when r = 3.76 cm 4.4 Assess Your Understanding (page 252) c F (a) 5x - x or x 6; - 1, 12 ∪ 12, q (b) 5x x … - or … x … 6; - q , - ∪ 1, (a) 5x x - or - x or x 6; - q , - 12 ∪ - 1, 12 ∪ 13, q (b) 5x … x or x … 6; 1, 22 ∪ 12, 5x x or x 6; - q , 02 ∪ 10, 32 11 5x͉ x 6; 11, q 13 5x x - or x 6; - q , - 42 ∪ 11, q 15 5x - x - or x 6; - 4, - 12 ∪ 10, q 17 5x x … - or x 6; - q , - 2] ∪ 11, q 19 5x x 6; 15, q 21 5x x 6; 14, q 23 5x x - 6; 1- q , - 52 25 5x x … - or - … x … - 6; 1- q , ∪ -2, - 14 27 5x - x or x 6; - 3, 02 ∪ 11, q 29 5x - x or x 6; - 3, 02 ∪ 10, 32 31 5x x 6; 11, q 33 5x x - or x 6; - q , - 12 ∪ 11, q 35 5x x … - or x … 6; - q , - ∪ 11, 37 5x x - or x 6; - q , - 22 ∪ 12, q 39 5x x 6; - q , 22 41 5x - … x - 6; - 8, - 22 43 5x - x - or x 6; - 7, - 12 ∪ 13, q 2 45 5x x - or … x or x Ú 6; - q , - 12 ∪ 30, 12 ∪ 2, q 47 e x͉ x or x f ; a - 1, b ∪ 12, q 3 1 49 5x - x or x 6; - 1, 32 ∪ 15, q 51 e x ` x … - or x Ú f ; - q , - 4 ∪ c , q b 53 5x x or x Ú ; 2 2 3 - q , 32 ∪ [7, q 55 5x x 6 ; - q , 22 57 e x ` x - or x f ; a - q , - b ∪ a0, b 59 5x x … - or … x … ; 3 - q , - ∪ 0, y y y ؍x ؉3 61 (a) 65 5x x ; 14, q 63 (a) 15 4, 10 10 28 67 5x x … - or x Ú ; - q , - ∪ 32, q , (؊6, 0) 10 , y؍1 ,؊ (؊4, 0) (1, 0) 10 x x؍2 (b) - q , - ∪ 1, 22 ∪ 12, q 71 y f (x) ؍x4 ؊ 2.5 (؊1, 0) (1, 0) 2.5 x (0, 2) 10 x (؊1, 0) x ؍؊2 (b) - 4, - 22 ∪ - 1, 32 ∪ 13, q 73 y g(x) ؍3x 32 (2, 12) (؊2, 12) 84 (0, - 4), (0, 4), (9, 0) 85 x2 - x - 4 86 ( - q , q ) 87 (0, 4), (1.33, 2.81) 2.5 x g(x) ؍؊2 x2 ؉ f 1x2 … g 1x2 if - … x … Historical Problems (page 266) f (x) ؍x4 ؊ f 1x2 … g 1x2 if - … x … 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 b2 2b3 bc x3 + ac bx + a + db 27 3 b 2b bc Let p = c and q = + d Then x3 + px + q 27 ax - Z03_SULL1772_10_GE_APPB_ANS.indd 1071 = = 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 = = 02/05/17 2:08 PM 1072 Answers Historical Problems 3HK = - p K = - p H3 + a - H3 - b 3H p3 H + K3 = - q K3 = - q - H p 3H = -q = -q 27H 27H - p3 = - 27qH 27H + 27qH - p3 = H3 = H3 = H3 = H = -q { { -q C 27 q C 22 127 2 q C4 + Choose the positive root for now K3 = + q B p + -q - -q -q q2 C4 + + q q2 C4 + p3 27 R p3 27 C - A + p3 27 p3 p3 -q q2 -q q2 + + + + 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 = x = 1272p 22 127 2 27 + K = x = H + K # 27 K3 = - q - J - 27q { 127q2 - 1272 - p3 -q p3 27 4.5 Assess Your Understanding (page 266) 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 = 0; yes 19 R = f a - b = 2; no 21 7; or positive; or negative 23 5; positive; negative 25 3; positive; or negative 27 4; positive; negative 29 5; or positive; negative 31 6; positive; negative 1 1 1 3 33 {1, { 35 {1, { 37 {1, {2, { , { , { , { 39 {1, {2, { , { , {3, { , { , {6 2 3 4 1 5 10 41 {1, { , {2, { , {3, {6, {9, {18 43 {1, { , { , { , {2, { , {5, { , { , { , {10, { 3 3 1 45 - 3, - 1, 2; f 1x2 = 1x + 32 1x + 12 1x - 22 47 - ; f 1x2 = ax + b 1x2 + 12 49 - 2, 25, - 25; f 1x2 = 1x + 22 1x - 252 1x + 252 2 1 51 - , 1, 22, - 22; f 1x2 = ax + b 1x - 12 1x - 222 1x + 222 53 2, multiplicity 2; - 2, - 1; f 1x2 = 1x + 22 1x + 12 1x - 22 2 1 1 55 - , - ; f 1x2 = ax + b 1x + 12 1x2 + 22 57 - 1, 59 e f 61 e - , 2, f 63 51 65 e f 67 e - 4, - , f 3 2 2 69 LB = - 2; UB = 2 71 LB = - 1; UB = 1 73 LB = - 3; UB = 2 75 LB = - 1; UB = 1 77 LB = - 2; UB = 79 f 102 = - 1; f 112 = 10 81 f - 32 = - 42; f - 22 = 5 83 f 11.72 = 0.35627; f 11.82 = - 1.021 85 r = - 0.60 87 r = - 2.17 89 r = 0.70 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 (؊1, 0) (؊2, 0) (؊1.5, ؊1.5625) 97 y (3, 24) 103 (2, 12) (0, 2) (1, 0) x (؊1, 0) x ,0 (0, ؊1) ͙2 ؊ ,0 y (0, 2) (؊1, ؊9) 99 y 2.5 (1, 0) 2.5 x y ؊ ,0 (0, ؊2) ͙2 ,0 (2, 0) x (1, ؊3) ,0 2.5 x (0, ؊2) 111 If f 1x2 = xn - c n, then f 1c2 = c n - c n = 0, so x - c is a factor of f 117 All the potential rational zeros are integers, so r either is an integer or is not a rational zero (and is therefore irrational) 105 - 8, - 4, - is not a potential rational zero. 123 No; by the Rational Zeros Theorem, is not a potential rational zero 3 124 ( - 1, 13) 125 f (x) = - 3(x - 5)2 + 71 126 10, - 232, 10, 232, 14, 02 127 - 3, 22 and 15, q 121 No; by the Rational Zeros Theorem, 4.6 Assess Your Understanding (page 274) one - 4i T F + i - i 11 - i 13 + i, i 15 - i, + 2i, - - i 17 f 1x2 = x4 - 14x3 + 77x2 - 200x + 208; a = 1 19 f 1x2 = x6 - 12x5 + 55x4 - 120x3 + 139x2 - 108x + 85; a = 1 21 f 1x2 = x5 - 5x4 + 11x3 - 13x2 + 8x - 2; a = 1 23 5i, - 3 25 - 3i, - 2, 27 + 2i, - 2, 5 29 - 3i, - 3, , Z03_SULL1772_10_GE_APPB_ANS.indd 1072 02/05/17 2:08 PM Answers Review Exercises 1073 31 - 1, 1, - i, i; f 1x2 = 1x + 12 1x - 12 1x + i2 1x - i2 33 2, - 2i, + 2i; f 1x2 = 1x - 22 1x - + 2i2 1x - - 2i2 35 - 3i, - 2i, 2i, 3i; f 1x2 = 1x + 3i2 1x + 2i2 1x - 2i2 1x - 3i2 1 37 - 7, 4, - 3i, 3i; f 1x2 = 1x + 72 1x - 42 1x + 3i2 1x - 3i2 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 y 49 50 - 22 51 6x3 - 13x2 - 13x + 20 52 perpendicular 41 130 43 (a) f 1x2 = 1x2 - 22x + 12 1x2 + 22x + 12 (b) - 10 x −2 −5 Review Exercises (page 277) Neither Polynomial of degree y (؊2, 0) (0, 8) (2, ؊1) (0, 3) 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 18 x (0, ؊1) x (؊4, ؊8) y (1, 0) 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) (؊4, 0) (؊5, ؊49) (3, 7) (2, 0) 10 x (1, 2) (2, 3) x 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) (2, 0) x (0, 0) (؊1, 6) ؊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 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) (3, ؕ) Number Chosen ؊2 Value of R R(؊2) ؍5 – ؍2 R(1) ؍؊4 R(4) Location of Graph Above x-axis Below x-axis Above x-axis Point on Graph (1, ؊4) (؊2, 5) Z03_SULL1772_10_GE_APPB_ANS.indd 1073 (4, ) 1– y (؊2, 5) ؊10 10 4, y؍2 (3, 0) x (1, ؊4) x؍0 02/05/17 2:08 PM 1074 Answers Review Exercises 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– 3 (؊2, 0) x ؊3, ؊ 15 (3, ) (1, ؊3) 5– (1, ؊3) 3, 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 ؊ –2 Value of R ( ) –– ؍ R(؊4) R Location of Graph Above x-axis Point on Graph ( ؊4, –– ؊ –2 ) ؍؊–– 11 Below x-axis (؊ 5– 2, ) ؊–– 11 (؊2, 2) (2, 3) (3, ؕ) 5– R (0) ؍1 R Above x-axis Below x-axis (0, 1) ( ( ) ؍؊ 5– 5– 2, ؊4, R (4) –– ؍ Above x-axis ؊ ( ) ) 11 ؊ –– 4, –– y؍1 x (2, 0) (؊3, 0) 11 –– 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, 0) (؊ؕ, ؊2) Number Chosen ؊3 Value of F F(؊3) ؍؊–– 27 Location of Graph Below x-axis Point on Graph (؊3, ؊ ) 27 –– (0, 2) ؊1, (0, 0) ؊1 F(؊1 ) ؍1–3 F(1) ؍؊1–3 –– F(3) ؍27 Above x-axis Below x-axis Above x-axis (؊1, ) 1– 27 – 1– (؊ؕ, 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– 1– , 1– (؊1, 2) Number Chosen ؊3 ؊ –2 Value of G G (؊3) –؍2 Location of Graph Above x-axis Point on Graph (؊3, ) 1– G (0) ؍2 G (3) – ؍4 Below x-axis Above x-axis ( ) 3– 2, ) ؊1 Above x-axis (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, (2, ؕ) G ؊ –2 ؍؊1 (؊ x + 3 y-intercept: 2; x-intercept: - x + b 5 Horizontal asymptote: y = 1, not intersected ؊1 (؊2, ؊1) x x؍1 ; domain: 5x x ≠ - 1, x ≠ 2 In lowest terms, G1x2 = (؊ؕ, ؊2) x؍2 1 , 2 (2, 32) ؊2 Interval 10 x (2, 32) 32 (0, 0) Above x-axis ) 4 Vertical asymptote: x = - 1; hole at a2, 6 y 40 ؊2, R(2) ؍32 Above x-axis Location of Graph Above x-axis Point on Graph ( )؍ y؍x 27 ؊3, ؊ 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 1, ؊ (3, ) (1 , ؊ ) 3, y (2, ؕ) 23 5x x or x 6; - q , 12 ∪ 12, q 2 ؊1 1 1 28 f 142 = 47,105 30 positive; or negative 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 Z03_SULL1772_10_GE_APPB_ANS.indd 1074 02/05/17 2:08 PM Answers Cumulative Review 1075 37 lb: - 2; ub: 3 38 lb: - 3; ub: 5 40 f 102 = - 1; f 112 = 1 41 1.52 42 0.93 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 Chapter Test (page 279) (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) (g) (e) Crosses at - 5, - , (f) y = 2x3 ؊ ,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 279) 126 5x x … or x Ú ; - q , or 1, q 5x - x 6; - 1, 42 ؊1 f 1x2 = - 3x + y (؊1, 4) 5 x 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) (1, 1) x Z03_SULL1772_10_GE_APPB_ANS.indd 1075 3 f; c , q b 2 y = 2x - y (3, 5) x y 10 (؊1, ؊1) (2, 8) (1, 1) 10 x (؊2, ؊8) 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) x - 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 02/05/17 2:08 PM 1076 16 Answers Cumulative Review 17 y (0, 7) 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 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 1؉ ,0 x (1, ؊1) x؍1 22 y (؊1, 5) (؊2, 2) (2, 5) (0, 2) x (0, 1) x (2, ؊2) (؊3, ؊5) 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 (d) Range: 5y y 6 or - q , 52 Chapter 5 Exponential and Logarithmic Functions 5.1 Assess Your Understanding (page 288) composite function; f(g(x)) F c a F (a) 5 (b) 11 (c) 0 (d) 0 (e) 1 (f) 7 11 (a) 3 (b) 4 (c) 1 (d) - 1 13 (a) 98 (b) 49 (c) 4 (d) 4 15 (a) 4418 (b) - 191 (c) 8 (d) - 2 17 (a) 213 (b) 23 (c) 22 + 1 (d) 0 19 (a) (b) 17 110 12 - 2 (c) 1 (d) 21 (a) (b) (c) 1 (d) 23 (a) (f ∘ g)(x) = 6x + 3; all real numbers (b) (g ∘ f)(x) = 6x + 9; all real numbers 25 (c) (f ∘ f)(x) = 4x + 9; all real numbers (d) (g ∘ g)(x) = 9x; all real numbers 25 (a) (f ∘ g)(x) = x2 + 5; All real numbers (b) (g ∘ f)(x) = x2 + 2x + 5; All real numbers (c) (f ∘ f)(x) = x + 2; All real numbers (d) (g ∘ g)(x) = x4 + 8x2 + 20; 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) 2 ; e x͉ x ≠ - ; x ≠ f (c) (f ∘ f)(x) = ; 5x͉ x ≠ 1, x ≠ (d) (g ∘ g)(x) = x; 5x͉ x ≠ 31 (a) (f ∘ g)(x) = - x x + 3x 2(x + 3) x ; e x͉ x ≠ - 3, x ≠ - f (d) (g ∘ g)(x) = x; 5x͉ x ≠ (b) (g ∘ f)(x) = ; 5x͉ x ≠ - 3, x ≠ (c) (f ∘ f)(x) = x 4x + 33 (a) (f ∘ g)(x) = - 2x - 1; e x͉ x … - f (b) (g ∘ f)(x) = - 2x - 2; 5x͉ x Ú (c) (f ∘ f)(x) = 2x - - 2; 5x͉ x Ú 6 (d) (g ∘ g)(x) = 4x - 1; All real numbers 35 (a) (f ∘ g)(x) = x + 2; 5x͉ x Ú (b) (g ∘ f)(x) = 2x2 + 2; All real numbers 13 14 (c) (f ∘ f)(x) = x4 + 8x2 + 20; All real numbers (d) (g ∘ g)(x) = 2x - - 2; 5x͉ x Ú 6 37 (a) (f ∘ g)(x) = ; ex ` x ≠ , x ≠ f 3x - 14 6x - 9x - 16 33 (b) (g ∘ f)(x) = ; 5x͉ x ≠ 2; x ≠ (c) (f ∘ f)(x) = x; 5x͉ x ≠ (d) (g ∘ g)(x) = ; ex ` x ≠ , x ≠ f x - 8x - 33 1 39 ( f ∘ g)(x) = f(g(x)) = f a x b = a x b = x; (g ∘ f )(x) = g( f (x)) = g(2x) = (2x) = x 2 41 (f ∘ g)(x) = f(g(x)) = f(x - 5) = x - + = x; (g ∘ f)(x) = g(f(x)) = g(x + 5) = x + - = x 1 1 43 ( f ∘ g)(x) = f(g(x)) = f a (4 - x) b = - a (4 - x) b = - + x = x; (g ∘ f)(x) = g(f(x)) = g(4 - 3x) = (4 - (4 - 3x)) = (3x) = x 3 3 1 x 1 x # # 45 ( f ∘ g)(x) = f(g(x)) = f a b = = = x; (g ∘ f)(x) = g(ƒ(x)) = g a b = = = x x x x x 47 f(x) = x ; g(x) = 2x + (Other answers are possible.) 49 f(x) = 2x; g(x) = - x (Other answers are possible.) 51 f(x) = ͉ x͉ ; g(x) = 2x2 + (Other answers are possible.) 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 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 - 5, - 3, 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) Z03_SULL1772_10_GE_APPB_ANS.indd 1076 05/05/17 11:58 AM Answers Section 5.2 1077 5.2 Assess Your Understanding (page 299) 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 Not one-to-one 27 29 Annual Rainfall (inches) Unemployment Rate State 11% 5.5% 5.1% 6.3% Virginia Nevada Tennessee Texas Location Atlanta, Georgia Boston, Massachusetts Las Vegas, Nevada Miami, Florida Los Angeles, California 49.7 43.8 4.2 61.9 12.8 Domain: {5.0,, 8.5,, 6.7,, 5.5,} Range: {Virginia, Nevada, Tennessee, Texas} 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, 33 - 8, - 22, - 1, - 12, 10, 02, 11, 12, 18, 22 Domain: - 8, - 1, 0, 1, Range: - 2, - 1, 0, 1, 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 1 37 f 1g 1x2 = f a x - b = a x - b + = x - + = x 2 g 1f 1x2 = g 12x + 62 = 12x + 62 - = x + - = x 41 ƒ 1g 1x2 = ƒ 1x2 = x g 1ƒ 1x2 = g 1x2 = x 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 3x + - - 2x 3x + 43 f 1g 1x2 = f a b = - 2x a 45 3x + - b 11 - 2x2 - 2x = 3x + 3x + 2a b + a2 a b + b 11 - 2x2 - 2x - 2x 3x + - 11 - 2x2 3x + - + 10x 13x = = x, x ≠ = = 13x + 52 + 11 - 2x2 6x + 10 + - 6x 13 x - g 1f 1x2 = g a b = 2x + 3a x - b + 2x + = f (x) ؍؊4x 5 ؊5 ؊5 x f ؊1(x) ؍؊ x f ؊1 y (2, 1) 2.5 x (؊2, ؊2) (1, 0) (؊1, 1) ؊2 ؊2 (0, ؊1) 49 y y؍x x f ؊1 (1, ؊2) y؍x 12x + 32 - 1x - 52 x - b - 2a 2x + 3x - 15 + 10x + 15 13x = = = x, x ≠ 2x + - 2x + 10 13 1 f(f -1(x)) = f a - x b = - a - x b = x 4 f -1(f(x)) = f -1( - 4x) = - ( - 4x) = 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) 47 y؍x f ؊1 1x - 52 + 12x + 32 51 (a) f -1(x) = - x y 2.5 f ؊1 x ؊2 ؊2 x x x -1 f(f (x)) = f a - b = a - b + 4 = (x - 2) + = x 4x + f -1(f(x)) = f -1(4x + 2) = 1 = ax + b = x 2 -1 (b) Domain of f = Range of f = All real numbers; 53 (a) f -1 1x2 = Range of f = Domain of f -1 = All real numbers y؍x y (c) f (x) ؍4x ؉ x f ؊1(x) ؍ Z03_SULL1772_10_GE_APPB_ANS.indd 1077 x ؊ 02/05/17 2:08 PM ... respect to the origin, replace x by - x and y by - y -y = 41 - x2 - x2 + 4x2 x2 + 4x2 y = - x + -y = Replace x by - x and y by - y Simplify Multiply both sides by - Since the result is not equivalent... Authorized adaptation from the United States edition, entitled Precalculus, 10th edition, ISBN 978-0-321-97907-0, by Michael Sullivan published by Pearson Education © 2016 All rights reserved No part... x-Axis: Replacing y by - y yields - y = , which is not equivalent to y = x x 1 y-Axis: Replacing x by - x yields y = = - , which is not equivalent to -x x y = x Origin: Replacing x by - x and y by - y