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TI-Nspire CAS OS3 and Casio ClassPad version ESSENTIAL MICHAEL EVANSKAY LIPSON PETER JONES SUE AVERY Mathematical Methods 3 & 4 CAS ENHANCED CAS calculator material prepared in collaboration with Jan Honnens David Hibbard Russell Brown ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans, Kay Lipson, Peter Jones, Sue Avery 2012 Cambridge University Press

P1: FXS/ABE P2: FXS 0521665175agg.xml CUAU030-EVANS January 3, 1970 15:16 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.edu.au Information on this title: www.cambridge.org/9781107676855 CMichael Evans, Kay Lipson, Peter Jones & Sue Avery 2011 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2005 TIN/CP version 2009 Enhanced version 2011 Reprinted 2012, 2013, 2014 Typeset by Aptara Corp. Cover designed by Marta White Printed in Singapore by C.O.S. Printers Pte Ltd. A Cataloguing-in-Publication entry is available from the catalogue of the National Library of Australia at www.nla.gov.au ISBN 978-1-107-67685-5 Paperback ISBN 978-1-139-28463-9 Electronic version Additional resources for this publication at www.cambridge.edu.au/GO Reproduction and communication for educational purposes The Australian Copyright Act 1968 (the Act) allows a maximum of one chapter or 10% of the pages of this publication, whichever is the greater, to be reproduced and/or communicated by any educational institution for its educational purposes provided that the educational institution (or the body that administers it) has given a remuneration notice to Copyright Agency Limited (CAL) under the Act. For details of the CAL licence for educational institutions contact: Copyright Agency Limited Level 15, 233 Castlereagh Street Sydney NSW 2000 Telephone: (02) 9394 7600 Facsimile: (02) 9394 7601 Email: info@copyright.com.au Reproduction and communication for other purposes Except as permitted under the Act (for example a fair dealing for the purposes of study, research, criticism or review) no part of this publication may be reproduced, stored in a retrieval system, communicated or transmitted in any form or by any means without prior written permission. All inquiries should be made to the publisher at the address above. Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables and other factual information given in this work is correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter. ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. © Michael Evans, Kay Lipson, Peter Jones, Sue Avery 2012 Cambridge University Press

P1: FXS/ABE P2: FXS 0521665175agg.xml CUAU030-EVANS September 1, 2011 9:28 Contents Introductionx CHAPTER 1— Functions and relations 1 1.1 Set notation 1 1.2 Identifying and describing relations and functions 5 1.3 Types of functions and maximal domains 14 1.4 The modulus function 20 1.5 Sums and products of functions 24 1.6 Composite functions 25 1.7 Inverse functions 31 1.8 Applications 36 Chapter summary 39 Multiple-choice questions 40 Short-answer questions (technology-free) 41 Extended-response questions 42 CHAPTER 2— Revising linear functions and matrices 45 2.1 Linear equations 45 2.2 Linear literal equations and simultaneous linear literal equations 47 2.3 Linear coordinate geometry 49 2.4 Applications of linear functions 54 2.5 Review of matrix arithmetic 56 2.6 Solving systems of linear simultaneous equations in two variables 64 2.7 Simultaneous linear equations with more than two variables 69 Chapter summary 76 Multiple-choice questions 77 iii ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. © Michael Evans, Kay Lipson, Peter Jones, Sue Avery 2012 Cambridge University Press

P1: FXS/ABE P2: FXS 0521665175agg.xml CUAU030-EVANS September 1, 2011 9:28 iv Contents Short-answer questions (technology-free) 78 Extended-response questions 79 CHAPTER 3— Families of functions 81 3.1 Functions with rule f(x)= xn 81 3.2 Dilations 86 3.3 Reflections 90 3.4 Translations 92 3.5 Combinations of transformations 95 3.6 Determining transformations to sketch graphs 99 3.7 Using matrices for transformations 104 3.8 Determining the rule for a function of a graph 110 3.9 Addition of ordinates 112 3.10 Graphing inverse functions 113 Chapter summary 119 Multiple-choice questions 122 Short-answer questions (technology-free) 124 Extended-response questions 125 CHAPTER 4— Polynomial functions 128 4.1 Polynomials 128 4.2 Quadratic functions 136 4.3 Determining the rule for a parabola 141 4.4 Functions of the form f:R→ R,f(x)= a(x+ h)n+ k, where nis a natural number 144 4.5 The general cubic function 149 4.6 Polynomials of higher degree 153 4.7 Determining rules for the graphs of polynomials 156 4.8 Solution of literal equations and systems of equations 160 Chapter summary 167 Multiple-choice questions 167 Short-answer questions (technology-free) 169 Extended-response questions 170 CHAPTER 5— Exponential and logarithmic functions 174 5.1 Exponential functions 174 5.2 The exponential function, f(x)= ex 180 5.3 Exponential equations 182 5.4 Logarithmic functions 184 ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. © Michael Evans, Kay Lipson, Peter Jones, Sue Avery 2012 Cambridge University Press

P1: FXS/ABE P2: FXS 0521665175agg.xml CUAU030-EVANS September 1, 2011 9:28 Contents v 5.5Determining rules for graphs of exponential and logarithmic functions 191 5.6 Change of base and solution of exponential equations 195 5.7 Inverses 198 5.8 Exponential growth and decay 202 Chapter summary 205 Multiple-choice questions 206 Short-answer questions (technology-free) 207 Extended-response questions 208 CHAPTER 6— Circular functions 211 6.1 Review of circular (trigonometric) functions 211 6.2 Graphs of sine and cosine 223 6.3 Transformations applied to graphs of y=sin xand y=cos x 224 6.4 Addition of ordinates 233 6.5 Determining the rule for graphs of circular functions 234 6.6 The function tan 236 6.7 General solution of circular function equations 242 6.8 Identities 247 6.9 Applications of circular functions 250 Chapter summary 253 Multiple-choice questions 254 Short-answer questions (technology-free) 256 Extended-response questions 256 CHAPTER 7— Functions revisited 260 7.1 Operations on functions 260 7.2 Inverse relations 265 7.3 Sums and products of functions and addition of ordinates 268 7.4 Identities with function notation 270 7.5 Families of functions and solving literal equations 272 Chapter summary 278 Multiple-choice questions 278 Short-answer questions (technology-free) 279 Extended-response questions 280 CHAPTER 8— Revision of Chapters 1–7 282 8.1 Multiple-choice questions 282 8.2 Extended-response questions 292 ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. © Michael Evans, Kay Lipson, Peter Jones, Sue Avery 2012 Cambridge University Press

P1: FXS/ABE P2: FXS 0521665175agg.xml CUAU030-EVANS September 1, 2011 9:28 vi Contents CHAPTER 9— Differentiation of polynomials, power functions and rational functions 296 9.1 The gradient of a curve at a point 296 9.2 The derived function 300 9.3 Differentiating xnwhere nis a negative integer 308 9.4 The chain rule 311 9.5 Differentiating rational powers  xpq 315 9.6 Product rule 317 9.7 Quotient rule 320 9.8 The graph of the gradient function 322 9.9 Review of limits and continuity 328 9.10 Differentiability 333 9.11 Miscellaneous exercises 339 Chapter summary 342 Multiple-choice questions 343 Short-answer questions (technology-free) 344 Extended-response questions 345 CHAPTER 10— Applications of differentiation 347 10.1 Tangents and normals 347 10.2 Angles between curves 351 10.3 Linear approximation 354 10.4 Stationary points 358 10.5 Types of stationary points 362 10.6 Absolute maxima and minima 370 10.7 Maxima and minima problems 374 10.8 Rates of change 379 10.9 Related rates of change 382 10.10 Families of functions 386 Chapter summary 391 Multiple-choice questions 391 Short-answer questions (technology-free) 393 Extended-response questions 394 CHAPTER 11— Differentiation of transcendental functions 402 11.1 Differentiation of ex 402 11.2 Differentiation of the natural logarithm function 406 11.3 Applications of differentiation of exponential and logarithmic functions 409 11.4 Derivatives of circular functions 416 ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. © Michael Evans, Kay Lipson, Peter Jones, Sue Avery 2012 Cambridge University Press

P1: FXS/ABE P2: FXS 0521665175agg.xml CUAU030-EVANS September 1, 2011 9:28 Contents vii 11.5Applications of derivatives of circular functions 421 11.6 Miscellaneous exercises 426 11.7 Applications of transcendental functions 427 Chapter summary 432 Multiple-choice questions 432 Short-answer questions (technology-free) 433 Extended-response questions 434 CHAPTER 12— Integration 441 12.1 Approximations leading to the definite integral 441 12.2 Antidifferentiation 447 12.3 Antidifferentiation of ( ax + b)r 451 12.4 The antiderivative of ekx 454 12.5 The fundamental theorem of calculus and the definite integral 456 12.6 Area under a curve 460 12.7 Integration of circular functions 464 12.8 Miscellaneous exercises 466 12.9 Area of a region between two curves 471 12.10 Applications of integration 475 12.11 The fundamental theorem of calculus 482 Chapter summary 486 Multiple-choice questions 487 Short-answer questions (technology-free) 488 Extended-response questions 491 CHAPTER 13— Revision of Chapters 9–12 496 13.1 Multiple-choice questions 496 13.2 Extended-response questions 504 CHAPTER 14— Discrete random variables and their probability distributions 509 14.1 Review of probability 509 14.2 Discrete random variables 519 14.3 Discrete probability distributions 521 14.4 Measures of centre and variability for a discrete random variable 524 Chapter summary 534 Multiple-choice questions 536 Short-answer questions (technology-free) 537 Extended-response questions 538 ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. © Michael Evans, Kay Lipson, Peter Jones, Sue Avery 2012 Cambridge University Press

P1: FXS/ABE P2: FXS 0521665175agg.xml CUAU030-EVANS September 1, 2011 9:28 viii Contents CHAPTER 15— The binomial distribution 542 15.1 Bernoulli sequences and the binomial probability distribution 542 15.2 The graph of the binomial probability distribution 549 15.3 Expectation and variance 552 15.4 Using the CAS calculator to find the sample size 555 Chapter summary 558 Multiple-choice questions 559 Short-answer questions (technology-free) 560 Extended-response questions 561 CHAPTER 16— Markov chains 564 16.1 Using matrices to represent conditional probability 564 16.2 Markov chains 569 16.3 Steady state of a Markov chain 577 16.4 Comparing run length for Bernoulli sequences and Markov chains 584 Chapter summary 589 Multiple-choice questions 590 Short-answer questions (technology-free) 591 Extended-response questions 592 CHAPTER 17— Continuous random variables and their probability distributions 594 17.1 Continuous random variables 594 17.2 Cumulative distribution functions 604 17.3 Mean, median and mode for a continuous random variable 607 17.4 Measures of spread 616 17.5 Properties of mean and variance 621 Chapter summary 626 Multiple-choice questions 627 Short-answer questions (technology-free) 629 Extended-response questions 631 CHAPTER 18— The normal distribution 633 18.1 The normal distribution 633 18.2 Standardisation and the 68–95–99.7% rule 639 18.3 Determining normal probabilities 644 ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. © Michael Evans, Kay Lipson, Peter Jones, Sue Avery 2012 Cambridge University Press

P1: FXS/ABE P2: FXS 0521665175agg.xml CUAU030-EVANS September 1, 2011 9:28 Contents ix 18.4Solving problems using the normal distribution 650 Chapter summary 655 Multiple-choice questions 655 Short-answer questions (technology-free) 657 Extended-response questions 658 CHAPTER 19— Revision of Chapters 14–18 661 19.1 Multiple-choice questions 661 19.2 Extended-response questions 666 CHAPTER 20— Revision of Chapters 1–19 673 Glossary 697 Appendix A — Counting methods and the binomial theorem 705 A1 Counting methods 705 A2 Summation notation 708 A3 The binomial theorem 709 Appendix B — Computer Algebra System (TI-Nspire) 713 Appendix C — Computer Algebra System (ClassPad 330) 729 Answers 740 ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. © Michael Evans, Kay Lipson, Peter Jones, Sue Avery 2012 Cambridge University Press

P1: FXS/ABE P2: FXS 0521665175agg.xml CUAU030-EVANS September 1, 2011 9:28 Introduction This book provides a complete course for Mathematical Methods Units 3 and 4 CAS. It has been written as a teaching text, with understanding as its chief aim and with ample practice offered through the worked examples and exercises. All the work has been trialled in the classroom, and the approaches offered are based on classroom experience.The book contains three revision chapters. These provide multiple-choice questions and extended-response questions. Chapter 20 contains 42 extended-response questions which may be used for revision. Use of a CAS calculator has been included throughout the text and there is also an appendix that provides an introduction to the use of the calculator. The use of matrices to describe transformations, solve systems of linear equations and in the study of Markov sequences is fully integrated. The study of families of functions is also incorporated throughout the text. Extended-response questions that require a CAS calculator have been incorporated. These questions are indicated by the use of a CAS calculator icon. The TI-Nspire calculator instructions have been completed by Jan Honnens and the Casio ClassPad instructions have been completed by David Hibbard. The TI-Nspire instructions are written for operating system 3.0 but can be used with other versions. The Casio ClassPad instructions are written for operating system 3 or above. x ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. © Michael Evans, Kay Lipson, Peter Jones, Sue Avery 2012 Cambridge University Press

P1: FXS/ABE P2: FXS 0521665175agg.xml CUAU030-EVANS September 1, 2011 9:28 Author profiles Michael Evans now works at the Australian Mathematical Sciences Institute based at the University of Melbourne. He was Head of Mathematics at Scotch College for many years and has been heavily involved in curriculum development at both Victorian state and national levels. Michael is a highly experienced writer of mathematical texts and is a lead author on a number of the texts in the Essential VCE Mathematics series.Peter Jones has been an active supporter of school mathematics over many years. Peter is a highly experienced writer of mathematical texts and he is the lead author on two of the Essential Mathematics textbooks. His area of expertise is applied statistics. Professor Kay Lipson’s experience extends through teaching mathematics and statistics at both secondary and tertiary level, as well as extensive periods as a VCE examiner. At the time of publication she was the Academic Dean for Swinburne University Online. The late Sue Avery was an experienced VCE Mathematics teacher and a key contributor to the Essential VCE Mathematics series. She conducted student seminars on VCE preparation in Maths Methods and Specialist Maths and had also been involved in maths research work for the VCAA. xi ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. © Michael Evans, Kay Lipson, Peter Jones, Sue Avery 2012 Cambridge University Press

In each chapter you will find … Enhanced TI-Nspire and Casio ClassPad versions Enhanced TI-Nspire and Casio ClassPad versions a vibrant full colour text with a clear layout that makes maths more accessible for students ‘Using a calculator’ boxes within chapters explain how to do problems using the TI-Nspire and ClassPad calculators, and include screen shots to further assist students a wealth of worked examples that support theory explanations within chapters carefully graduated exercises that include a number of easier lead-in questions to provide students with a greater opportunity for immediate success chapter reviews that include multiple-choice, short-answer (technology-free) and extended- response questions chapter summaries at the end of each chapter provide students with a coherent overview TI-Nspire and Casio ClassPad appendices that provide step-by- step worked examples using a CAS calculator a comprehensive glossary of mathematical terms with page references to assist in the ‘openbook’ exam revision chapters to help consolidate student knowledge ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans, Kay Lipson, Peter Jones, Sue Avery 2012 Cambridge University Press

Explaining icons in the book ... What teachers and students will find in the digital resources ... Indicates that a skillsheet is available to provide further practice and examples in this area. If students are having difficulty they can approach their teacher who can access this material on the Teacher CD-ROM. Links to Teacher CD-ROM Live links to interactive files included in the online Interactive Textbook on Cambridge GO. Links to resources in the Interactive Textbook The Essential Mathematics Methods 3 & 4 Teacher CD- ROM contains a wealth of time-saving assessment and classroom resources including: modifiable chapter tests and answers containing multiple-choice and short-answer questions chapter review assignments with extended problems that can be given to students in class or can be completed at home printable versions of the multiple-choice questions from the Interactive Textbook print-ready skillsheets to revise the prerequisite knowledge and skills required for the chapter editable Exam Question Sets from which teachers can create their own exams. Updates to the Teacher CD-ROM are published on Cambridge GO Teacher CD-ROM The online Interactive Textbook is an HTML version formatted for navigation and reading on screen. Additional resources are hyperlinked from the textbook, and include interactive multiple-choice questions, drag- and-drop activities and technology applets such as PowerPoint and Excel activities. Interactive Textbook A downloadable PDF of the textbook is also available on Cambridge GO for students to use when they are unable to go online. Textbook PDF Additional resources ... The Essential Mathematics Methods 3 & 4 Solutions Supplement book provides solutions to the extended- response questions, highlighting the process as well as the answer. Updates for the enhanced version are published on Cambridge GO. Solutions Supplement Cambridge GO is the new home of the companion websites for the Essentials series for students and teachers. In addition to the resources detailed above, these pages provide updates and contact details. Websites www.cambridge.edu.au/go Ess Maths IN BOOK BROCHUR 1.qxd 8/23/11 8:02 PM Page v ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. © Michael Evans, Kay Lipson, Peter Jones, Sue Avery 2012 Cambridge University Press

This textbook is supported and enhanced by online resources... www.cambridge.edu.au/GO Digital resources and support material for schools. About the free online resources... Free additional student and teacher support resources are available online at Cambridge GO and include: • the PDF Textbook – a downloadable version of the student text, with note-taking and bookmarking enabled • extra material and activities • links to other resources. Available free for users of this textbook. Use the unique access code found in the front of this textbook to activate these resources. About Cambridge GO Interactive... Cambridge GO Interactive includes the Interactive Textbook, and is designed to make the online reading experience meaningful, from navigation to display. It also contains a range of extra features that enhance teaching and learning in a digital environment. Access the Interactive Textbook by purchasing a unique access code from your Educational Bookseller, or you may have already purchased Cambridge GO Interactive as a bundle with this printed textbook. The access code will be enclosed in a separate sealed \ pocket. Cambridge GO Interactive is available on a calendar year subscription. For a limited time only, access to this subscription has been included with the purchase of the enhanced version of the printed student text at no extra cost. You are not automatically entitled to receive any additional interactive content or updates that may be provided on Cambridge GO in the future. Preview online at: Interactive Maths VCE 256mmx190mm CG DBL page spread Interactive.indd \ 2 6/07/11 10:05 AM ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans, Kay Lipson, Peter Jones, Sue Avery 2012 Cambridge University Pres•

Access online resources today at www.cambridge.edu.au/GO Go to the My Resources page on Cambridge GO and access all of your resources anywhere, anytime.* * Technical specifications: You must be connected to the internet to activate your account and to use the Interactive Textbook. Some material, including the PDF Textbook, can be downloaded. To use the PDF Textbook you must have the latest version of Adobe Reader installed. 2. 3. 1. Log in to your existing Cambridge GO user account OR Create a new user account by visiting: www.cambridge.edu.au/GO/newuser • All of your Cambridge GO resources can be accessed through this account. • You can log in to your Cambridge GO account anywhere you can access the internet using the email address and password with which you’re registered. Activate Cambridge GO resources, including the PDF textbook, by entering the unique access code found in the front of this textbook. Activate the Interactive Textbook by entering the unique access code found in the separate sealed pocket. • Once you have activated your unique code on Cambridge GO, you don’t need to input your code again. You can just login to your account using the email address and password you registered with and you will find all of your resources. For more information or help, contact us on 03 8671 1400 or enquiries@cambridge.edu.au\ Interactive Maths VCE 256mmx190mm CG DBL page spread Interactive.indd \ 3 6/07/11 10:05 AM ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans, Kay Lipson, Peter Jones, Sue Avery 2012 Cambridge University Pres•

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 CHAPTER 1 Functions and relations Objectives To understand and use thenotation of sets, including the symbols ∈,⊆, ∩,∪,∅and \. To use the notation for sets of numbers. To understand the concept of relation. To understand the termsdomainandrange. To understand the concept of function. To understand the termone-to-one. To understand the terms implied domain,restriction of a function, hybrid function, and odd and even functions. To understand the modulus function. To understand and use sumsandproducts of functions. To define composite functions. To understand and find inverse functions. To apply a knowledge of functions to solving problems. In this chapter, notation that will be used throughout the book will be introduced. The language introduced in this chapter is necessary for expressing important mathematical ideas precisely. If you are working with a CAS calculator it is appropriate to work through the first sections of the appropriate Computer Algebra System Appendix. 1.1 Set notation Set notationis used widely in mathematics and in this book it is employed where appropriate. This section summarises much of the set notation you will need. Aset is a collection of objects. The objects that are in the set are known as the elementsor members of the set. If xis an element of a set Awe write x∈ A.This can also be read as ‘x is a member of the set A’or‘xbelongs to A’or‘xis in A’. The notation x/ ∈ Ameans xis not an element of A. For example: 2 / ∈ set of odd numbers. A set Bis called a subsetof a set Aif and only if x∈ Bimplies x∈ A. 1 ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 2 Essential Mathematical Methods3&4CAS To indicate thatBis a subset of A, we writeB⊆ A.This expression can also be read as ‘B is contained in A’or‘Acontains B’. The set of elements common to two sets Aand Bis called the intersection ofAand Band is denoted by A∩ B.Thus x∈ A∩ Bif and only if x∈ Aand x∈ B. If the sets Aand Bhave no elements in common, we say Aand Bare disjoint, and write A ∩ B=∅ . The set ∅is called the emptyset ornullset. The union of sets Aand B, written A∪ B, is the set of elements that are either in Aor in B. This does not exclude objects that are elements of both Aand B. Example 1 A={ 1,2,3,7}; B={ 3,4,5,6,7} Find: a A∩ B bA∪ B Solution a A∩ B= {3, 7} bA∪ B={ 1,2,3,4,5,6,7} Note: In this example, 3 ∈Aand 5 / ∈ Aand {2, 3} ⊆A. Finally, the set difference of two setsAand Bis denoted A\B, where: A \B ={ x:x ∈ A,x / ∈ B} e.g., for Aand Bin Example 1, A\B={1, 2} andB\A ={4, 5, 6} There will be a further discussion of set notation in Chapter 14, which will provide the additional notation necessary for the study of probability. Sets of numbers The elements of the set {1,2,3,4,... }are called the natural numbers. The set of natural numbers will be denoted by N. The elements of {...,−2,−1,0,1,2,... }are called integers. The set of integers will be denoted by Z. The numbers of the form p qwith pand qintegers, q= 0, are called rational numbers. The rational numbers may be characterised by the property that each rational number may be written as a terminating or recurring decimal. The set of rational numbers will be denoted by Q. The real numbers that are not rational numbers are called irrational(e.g.,and \b 2). The set of real numbers will be denoted by R. It is clear that N⊆ Z⊆ Q⊆ Rand this may be represented by the diagram: NZQR ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 Chapter 1 — Functions and relations 3 Note: {x :0 0,xrational} is the set of all positive rational numbers. {2n :n = 0,1,2,... }is the set of all even numbers. Among the most important subsets of Rare the intervals. The following is an exhaustive list of the various types of intervals and the standard notation for them. We suppose that aand b are real numbers and that a< b: ( a ,b )={ x:a < x< b} [a ,b ]={ x:a ≤ x≤ b} ( a ,b ]={ x:a < x≤ b} [a ,b )={ x:a ≤ x< b} ( a ,∞) ={x:x > a} [a ,∞) ={x:x ≥ a} (−∞, b)={ x:x < b} (−∞,b]={ x:x ≤ b} Inter v als may be represented by diagrams, as shown in Example 2. Example 2 Illustrate each of the following intervals of the real numbers on a number line: a [−2, 3] b(−3, 4] c(−∞, 5] d(−2, 4) e(−3, ∞) Solution –5 –4 –3 –2 –1 1 023456 a b c d e –5 –4 –3 –2 –1 1 023456 –5 –4 –3 –2 –1 1 023456 –5 –4 –3 –2 –1 1023456 –5 – 4 –3 –2 –1 1023456 The ‘closed’ circle indicates that the number is included. The ‘open’ circle indicates that the number is not included. The following are also subsets of the real numbers for which there are special notations: R +={x:x > 0} R −={ x:x < 0} R \{0} is the set of real numbers excluding 0. Z +={ x:x ∈ Z,x > 0} The cartesian plane is denoted by R 2whereR 2={(x , y): x ∈ R and y∈ R} ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 4 Essential Mathematical Methods3&4CAS Exercise1A 1For X= {2, 3, 5, 7, 9, 11}, Y= {7, 9, 15, 19, 23} andZ= {2, 7, 9, 15, 19}, find: a X∩Y bX∩Y ∩ Z cX∪Y dX\Y e Z\Y fX∩ Z g[−2, 8]∩X h(−3, 8]∩Y i (2, ∞) ∩Y j(3, ∞) ∪Y 2 For X= {a, b,c,d, e} and Y= {a, e,i,o, u), find: a X∩ Y bX∪ Y cX\Y dY\X 3 Fo r A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, B= {2, 4, 6, 8, 10} andC= {1, 3, 6, 9}, find: a B∩C bB\C cA\B d(A \B )∪ (A \C ) e A\( B ∩C) f(A \B )∩ (A \C ) gA\( B ∪C) hA∩ B∩C 4 Use the appropriate interval notation, i.e. [a ,b], (a,b ) etc., to describe each of the following sets: a {x :−3 ≤x< 1} b{x :−4

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 Chapter 1 — Functions and relations 5 1.2 Identifying and describing relations and functions Anordered pair, denoted (a, b), is a pair of elementsaand bin which ais considered to be the first element and bthe second. In this section, only ordered pairs of real numbers are considered. Two ordered pairs (a, b) and (c,d) are equal if a= cand b= d. A relation is a set of ordered pairs. The following are examples of relations: S ={ (1,1),(1,2),(3,4),(5,6)} T ={ (−3, 5),(4,12), (5,12), (7,−6)} Every relation determines two sets defined as follows: The domain of a relation Sis the set of all first elements of the ordered pairs in S. Therange of a relation Sis the set of all second elements of the ordered pairs in S. In the above examples: domain of S= {1, 3, 5}; range of S= {1, 2, 4, 6} domain of T= {−3, 4, 5, 7}; range of T= {5, 12, −6} A relation may be defined by a rule which pairs the elements in its domain and range. Thus the set {(x, y): y= x+ 1, x∈{ 1,2,3,4}} is the relation {(1,2),(2,3),(3,4),(4,5)} When the domain of a relation is not explicitly stated, it is understood to consist of all real numbers for which the defining rule has meaning. For example: S={ (x , y): y= x 2} is assumed to have domain Rand T ={ (x , y): y= \b x } is assumed to have domain [0, ∞). Example 3 Sketch the graph of each of the following relations and state the domain and range of each. a{(x, y): y= x 2} b{(x, y): y≤ x+ 1} c {(−2, −1),(−1,−1),(−1,1),(0,1),(1,−1)} d{(x, y): x 2+ y 2=1} e {(x, y): 2 x+ 3y = 6,x≥ 0} f{(x, y): y= 2x − 1, x∈ [−1, 2]} ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 6 Essential Mathematical Methods3&4CAS Solution 0 x y a b x –1 1 0 y Domain =R; range =R +∪{ 0} Domain=R; range =R x 0 –2 –1 –1 –2 12 1 2 y cd x 1 1 0 –1 –1 y Domain={−2,−1, 0,1}; range ={−1,1} Domain ={x:−1 ≤x≤ 1}; range ={y:−1 ≤y≤ 1} ef x 2 1 10 (0, 2) 23 y x –1 1 2 y (2, 3) (–1, –3) Domain =[0, ∞); range =(−∞, 2] Domain =[−1, 2]; range =[−3, 3] Sometimes the set notation is not used in the specification of a relation. For the above example: a is written as y= x 2 b is written as y≤ x+ 1 e is written as 2 x+ 3y = 6,x≥ 0 ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 Chapter 1 — Functions and relations 7 Afunction is a relation such that no two ordered pairs of the relation have the same first element. For instance, in Example 3, a,eand fare functions but b,cand dare not. Let Sbe a relation with domain D. A simple geometric test to determine if Sis a function is as follows. Consider the graph of S. If all vertical lines with equations x= a,a ∈ D, cut the graph of S only once, then Sis a function. For example, x 0 y x 0 y x2+ y 2=1 is not a function y= x 2is a function Functions are usually denoted by lower case letters such as f,g, h. The definition of a function tells us that for each xin the domain of fthere is a unique element, y, in the range such that (x, y)∈f. The element yis called the imageofxunder for the value offat xand is denoted by f(x) (read ‘f ofx’). If (x, y)∈f, then xis called a pre-image ofy. This gives an alternative way of writing functions. 1 For the function {(x, y): y= x 2}, write: f : R → R, f(x ) = x 2 2For the function {(x,y):y= 2x −1, x∈ [0, 4]} write: f :[0 ,4] → R, f(x ) = 2x − 1 3 For the function  (x , y): y= 1 x ,write: f : R \{0}→ R, f(x ) = 1 x If the domain is Rwe often just write the rule, for example in 1f(x) =x 2. Note that in using the notation f: X → Y,Xis the domain but Yis not necessarily the range. It is a set that contains the range and is called the codomain. With this notation for functions the domain of fis written as dom fand range of fas ran f. ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

8 Essential Mathematical Methods3&4CAS Using the TI-Nspire Function notation can be used with a CAS calculator. Use b >Actions>Defineto define the function f( x ) = 4x − 3. Type f( −3 ) followed by enterto evaluate f ( −3 ) . Type f( { 1, 2,3} ) followed by enterto evaluate f( 1 ) , f( 2 ) and f( 3 ) . Using the Casio ClassPad Function notation can be used with a CAS calculator. In tap Interactive–Define and enter the function name, variable and expression as shown. See page 10 for a screen showing the Define window. Enter f(−3) in the entry line and tap . In the entry line, type f({1,2,3}) to obtain the values of f(1), f(2) and f(3). Exa mple 4 If f(x ) = 2x 2+ x,find f(3), f(−2) and f(x − 1). Solu tion f(3) =2(3) 2+3= 21 f (−2) =2(−2) 2−2= 6 f (x − 1) =2(x− 1) 2+ x− 1 = 2(x 2− 2x + 1) +(x − 1) = 2x 2− 3x + 1 ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans, Kay Lipson, Peter Jones, Sue Avery 2012 Cambridge University Press

Chapter 1 — Functions and relations 9 Example 5 If f(x ) = 2x + 1,find f(−2) and f 1 a ,a = 0. Solu tion f(−2) =2(−2) +1=− 3 f  1 a = 2 1 a + 1= 2 a+ 1 Using the TI-Nspire Use b >Actions>Define to define the function f( x ) = 2x + 1. Type f( −2 ) followed by enterto evaluate f ( −2 ) . Type f 1 a followed by enterto evaluate f  1 a . Using the Casio ClassPad In tap Interactive–Define and enter the function name f, variablexand expression 2 x+ 1. Now complete f(−2) and f 1 a . Exa mple 6 Consider the function defined by f(x ) = 2x − 4 for all x∈ R. a Find the value of f(2), f(−1) and f(t ). bFor what values of tis f(t )= t? c For what values of xis f(x ) ≥ x? dFind the pre-image of 6. ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans, Kay Lipson, Peter Jones, Sue Avery 2012 Cambridge University Press

10 Essential Mathematical Methods3&4CAS Solution a f(2) =2(2) −4 = 0 f (−1) =2(−1) −4 =− 6 f (t )= 2t −4 b f(t )= t 2t −4= t t − 4= 0 ∴ t= 4 c f(x ) ≥ x 2 x − 4≥ x x − 4≥ 0 ∴ x≥ 4 d f(x ) = 6 2 x − 4= 6 x = 5 5 is the pre-image of 6. Using the TI-Nspire Use b>Actions>Define to define the function and b>Algebra> Solveto solve as shown. The symbol ≥can be found using /+= and select ≥or use /+b>Symbols. On the Clickpad (grey handheld) you can use /+> or>=. Using the Casio ClassPad Ta p Interactive–Define and enter the function name, variable and expression as shown. Enter and highlight f(x ) = x, tap Interactive–Equation/inequality– solve and ensure the variable is set as x. To enter the inequality, press k and look in the to find the ≥ symbol. ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans, Kay Lipson, Peter Jones, Sue Avery 2012 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 Chapter 1 — Functions and relations 11 Restriction of a function Consider the following functions: fx 0f (x) g x 0–1 1 g(x) h x 0 h(x) f(x ) = x 2,x ∈ Rg (x ) = x 2,−1 ≤x≤ 1 h(x ) = x 2,x ∈ R +∪{ 0} The different letters, f,gand h, used to name the functions, emphasise the fact that there are three different functions even though they each have the same rule. They are different because they are defined for different domains. We call gand hrestrictions of fsince their domains are subsets of the domain of f. Example 7 For each of the following, sketch the graph and state the range: af:[ −2, 4]→ R, f(x ) = 2x − 4 bg:( −1, 2]→ R,g (x ) = x 2 Solution a b x –4 02 (4, 4) (–2, –8) y x 0 (–1, 1) (2, 4) y Range =[−8, 4] Range=[0, 4] ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 12 Essential Mathematical Methods3&4CAS Exercise1B 1State the domain and range for the relations represented by each of the following graphs: y x x 1 0 –1 –2 1 2 2 y abc def x 0 12 1 2 y (– 2, 4) (3, 9) x (1, 2) (–3, – 6) y x –2 –1 1 2 0 y x 1 2 0 y 2 Sketch the graph of each of the following relations and state the domain and range of each: a{(x, y): y= x 2+1} b{(x, y): x 2+ y 2=9} c {(x, y): 3 x+ 12 y= 24, x≥ 0) dy= \b 2 x e {(x, y): y= 5− x,x ∈ [0, 5]} fy= x 2+2, x∈ [0, 4] g y= 3x − 2, x∈ [−1, 2] hy= 4− x 2 3 Which of the following relations are functions? State the domain and range for each. a {(−1, 1),(−1, 2),(1,2),(3,4),(2,3)} b {(−2, 0),(−1, −1),(0,3),(1,5),(2,−4)} c {(−1, 1),(−1, 2),(−2, −2),(2,4),(4,6)} d {(−1, 4),(0,4),(1,4),(2,4),(3,4)} e{(x,4): x∈ R} f {(2, y): y∈ Z} gy=− 2x + 4 h y≥ 3x + 2 i{x , y): x 2+ y 2=16} 4 Consider the function g(x ) = 3x 2− 2. a Find g(−2), g(4). bState the range of g. 5 Let f(x ) = 2x 2+ 4x and g(x ) = 2x 3+ 2x − 6. a Evaluate f(−1), f(2) and f(−3). bEvaluate g(−1), g(2) and g(3). c Express the following in terms of x: i f(−2 x) iif(x − 2) iiig(−2 x) ivg(x + 2) vg(x 2) ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 Chapter 1 — Functions and relations 13 6Consider the function f(x ) = 2x − 3. Find: a the image of 3 bthe pre-image of 11 c{x : f(x ) = 4x } 7 Consider the functions g(x ) = 6x + 7 and h(x ) = 3x − 2. Find: a {x :g (x ) = h(x )} b{x :g (x ) > h(x )} c{x :h (x ) = 0} 8 Rewrite each of the following using the f: X → Ynotation: a {(x, y): y= 2x + 3} b{(x, y): 3y + 4x = 12 } c {(x, y): y= 2x − 3, x≥ 0} dy= x 2−9, x∈ R e y= 5x − 3,0≤ x≤ 2 9 Sketch the graphs of each of the following and state the range of each: a y= x+ 1, x∈ [2, ∞) by=− x+ 1, x∈ [2, ∞) c y= 2x + 1, x∈ [−4, ∞) dy= 3x + 2, x∈ (−∞, 3) e y= x+ 1, x∈ (−∞, 3] fy= 3x − 1, x∈ [−2, 6] g y=− 3x − 1, x∈ [−5, −1] hy= 5x − 1, x∈ (−2, 4) 10 For f(x ) = 2x 2− 6x + 1 and g(x ) = 3− 2x : a Evaluate f(2), f(−3), f(−2). bEvaluate g(−2), g(1) and g(−3). c Express the following in terms of a: i f(a ) iif(a + 2) iiig(− a) ivg(2 a) v f(5 −a) vif(2 a) viig(a ) + f(a ) viii g(a ) − f(a ) 11 For f(x ) = 3x 2+ x− 2, find: a {x : f(x ) = 0} b{x : f(x ) = x} c{x : f(x ) =− 2} d {x : f(x ) > 0} e{x : f(x ) > x} f{x : f(x ) ≤− 2} 12 Forf(x ) = x 2+x, find: a f(−2) bf(2) cf(− a) in terms of adf(a ) + f(− a) in terms of a e f(a ) − f(− a) in terms of aff(a 2) in terms of a 13 Forg(x ) = 3x − 2, find: a {x :g (x ) = 4} b{x :g (x ) > 4} c{x :g (x ) = a} d {x :g (− x) = 6} e{x :g (2 x) = 4} f x: 1 g (x ) = 6 ,g (x ) = 0 14 Find the value of kfor each of the following if f(3) =3, where: a f(x ) = kx −1 bf(x ) = x 2−k cf(x ) = x 2+kx +1 d f(x ) = k x e f(x ) = kx 2 f f(x ) = 1− kx 2 15 Find the values of xfor which the given functions have the given value: a f(x ) = 5x − 4, f(x ) = 2 bf(x ) = 1 x, f(x ) = 5 c f(x ) = 1 x2, f(x ) = 9 d f(x ) = x+ 1 x, f(x ) = 2 e f(x ) = (x + 1)( x− 2), f(x ) = 0 ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 14 Essential Mathematical Methods3&4CAS 1.3 Types of functions and maximal domains One-to-one and many-to-one functions A functionfis said to be one-to-oneif fora,b ∈ dom f,a = b, then f(a ) = f(b ). In other words fis called one-to-one if every image under fhas a unique pre-image. The function f(x ) = 2x + 1 is a one-to-one function. The function f(x ) = x 2is not a one-to-one function as, for example, f(−3) =9 and f(3) =9; i.e., 9 does not have a unique pre-image. The function f(x ) = 5 is not a one-to-one function as there are infinitely many pre-images of 5. The function f(x ) = x 3is a one-to-one function. A geometric test for a function to be one-to-one is as follows. If for any a∈ ran fthe horizontal line, y= a, crosses the graph of fat only one point, the function is one-to-one. y = x 2 x 0y y= 2 x+ 1 x 0 y f(x)= 5 0 x y not one-to-one one-to-onenot one-to-one y= x 3 0 x y –330 3 x y one-to-one not one-to-one A function that is not one-to-one is many-to-one. ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 Chapter 1 — Functions and relations 15 Implied domains (maximal domains) If the domain is unspecified, then the domain is the largest subset ofRfor which the rule is defined. When the domain is not explicitly stated, it is implied by the rule. Thus for the function, f(x ) = \b x the implied domain (maximal domain) is [0, ∞). We write: f:[0 ,∞) →R, f(x ) = \b x Example 8 Find the implied domain of the functions with the following rules: a f(x ) = 2 2 x − 3 b g(x ) = \b 5 − x c h(x ) = \b x − 5+ \b 8 − x df(x ) = \b x2− 7x + 12 Solution a f(x ) is not defined when 2 x− 3= 0, i.e. when x= 3 2. Thus the implied domain is R\ 3 2 . b g(x ) is defined when 5 −x≥ 0, i.e. when x≤ 5. Thus the implied domain is (−∞, 5]. c h(x ) is defined when x− 5≥ 0 and 8 −x≥ 0, i.e. when x≥ 5 and x≤ 8. Thus the implied domain is [5, 8]. d f(x ) is defined when x 2−7x + 12 ≥0. x 2− 7x + 12 ≥0 is equivalent to ( x− 3)( x− 4) ≥0. Therefore, x≥ 4or x≤ 3. Thus the implied domain is (−∞, 3]∪[4, ∞). x 0 12345 y = x 2 – 7x + 12 y Hybrid functions Example 9 Sketch the graph of the function fgiven by: f (x ) = ⎧ ⎪ ⎪ \b ⎪ ⎪ ⎩ − x− 1 for x< 0 2 x − 1 for 0 ≤x≤ 1 1 2 x + 1 2 for x≥ 1 ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 16 Essential Mathematical Methods3&4CAS Solution x 01 1 2 3 –1 –1 –2 2 3 f(x) Functions like this, which have different rules for different subsets of the domain, are called hybrid functions. Odd and even functions Anodd function has the property that f(− x) =− f(x ). For example, f(x ) = x 3−xis an odd function since f(− x) = (− x) 3− (− x) =− x 3+x =− f(x ) x 1 –1 0 y 1 –1 0 x y y= f(x ) y= f(− x) An even function has the property that f(− x) = f(x ). For example, f(x ) = x 2−1 is an even function since f(− x) = (− x) 2− 1 = x 2−1 = f(x ) x y = x 2– 1 – 1 0 y The graphs of even functions are symmetrical about the y-axis. The properties of odd and even functions often facilitate the sketching of graphs. Exercise 1C 1State which of the following functions are one-to-one: a{(2, 3),(3,4),(5,4),(4,6)} b{(1, 2),(2,3),(3,4),(4,6)} c {(x, y): y= x 2+2} d{(x, y): y= 2x + 4} e f(x ) = 2− x 2 f y= x 2,x ≥ 1 ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 Chapter 1 — Functions and relations 17 2The following are graphs of relations. aState which are the graphs of a function. b State which are the graphs of a one-to-one function. x 0 1 2y i ii iii x 2 –2 0 y x 210 1 2 y iv vv i x 4 –4 0y x 0 y x 0 y vii viii x 0 y x 0 y 3 The graph of the relation {(x, y): y 2= x+ 2, x≥− 2} is shown. From this relation, form two functions and specify the range of each. x 0–1–2 y ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 18 Essential Mathematical Methods3&4CAS 4aDraw the graph of g:R → R,g (x ) = x 2+2. b By restricting the domain of g, form two one-to-one functions that have the same rule as g. 5 State the largest possible domain and range for the functions defined by the rule: a y= 4− x by= \b x cy= x 2−2 dy= \b 16 −x 2 e y= 1 x f y= 4− 3x 2 g y= \b x − 3 6 Each of the following is the rule of a function. In each case write down the implied domain and the range. a y= 3x + 2 by= x 2−2 cf(x ) = \b 9 − x 2 d g(x ) = 1 x − 1 7 Find the implied domain for each of the following rules: a f(x ) = 1 x − 3 b f(x ) = \b x2− 3 cg(x ) = \b x2+ 3 d h(x ) = \b x − 4+ \b 11 −x e f(x ) = x 2− 1 x+ 1 f h(x ) = \b x2− x− 2 g f(x ) = 1 ( x + 1)( x− 2) h h(x ) = x − 1 x+ 2 i f(x ) = \b x − 3x 2 j h(x ) = \b 25 −x 2 k f(x ) = \b x − 3+ \b 12 −x 8 Which of the following functions are odd, even or neither? a f(x ) = x 4 b f(x ) = x 5 c f(x ) = x 4−3x d f(x ) = x 4−3x 2 e f(x ) = x 5−2x 3 f f(x ) = x 4−2x 5 9a Sketch the graph of the function: f(x ) = ⎧ ⎪ \b ⎪ ⎩ −2 x− 2, x< 0 x − 2, 0≤ x< 2 3 x − 6, x≥ 2 b What is the range of f? 10 State the domain and range of the function for which the graph is shown. x 1 1 –1 –2 0 –3 3 2 3 y ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 Chapter 1 — Functions and relations 19 11State the domain and range of the function for which the graph is shown. x (4, 5) (1, 2) (–5, – 4)3 0 y 12 a Sketch the graph of the function with rule: f(x ) = ⎧ ⎪ \b ⎪ ⎩ 2 x + 60 0 5 − x −3≤x≤ 0 8 x< −3 b State the range of the function. 14 Given that f(x ) = ⎧ \b ⎩ 1 x , x> 3 2 x , x≤ 3 find: a f(−4) bf(0) cf(4) df(a + 3) in terms of a e f(2 a) in terms of aff(a − 3) in terms of a 15 Given that f(x ) = \b x − 1, x≥ 1 4, x< 1 find: a f(0) bf(3) cf(8) d f(a + 1) in terms of aef(a − 1) in terms of a 16 Sketch the graph of the function: g(x ) = ⎧ ⎪ ⎪ \b ⎪ ⎪ ⎩− x− 2, x< −1 x − 1 2 , −1≤x< 1 3 x − 3, x≥ 1 ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 20 Essential Mathematical Methods3&4CAS 17Specify the function illustrated by the graph. x –1 1 –2 –2–1 1 2 0 234 –3 –4 (–2, – 2) 1 2 3, y 1.4 The modulus function The modulus or absolute value of a real number xis denoted by |x|and is defined by: | x |= x ifx≥ 0 − x ifx< 0 It may also be defined as |x |= \b x2. For example, |5|=5 and|−5|=5. The function |x | has the following properties: |ab |=| a||b | a b = | a | |b | |a + b|≤| a|+| b|.If aand bare both non-negative or both non-positive, then equality holds. If a≥ 0,|x |≤ ais equivalent to −a ≤ x≤ a. If a≥ 0,|x − k|≤ ais equivalent to k− a≤ x≤ k+ a. Example 10 Evaluate each of the following: a i|−3 ×2| ii|−3|×| 2| b i −4 2 ii|−4| |2| c i|−6 +2| ii|−6|+| 2| Solution a i|−3 ×2|=|− 6|=6 ii|−3|×| 2|=3×2= 6 Note: |−3 ×2|=|− 3|×|2| b i −4 2 =|− 2|=2 ii|−4| |2| = 4 2= 2 Note: −4 2 = |−4| |2| c i|−6 +2|=|− 4|=4 ii|−6|+| 2|=6+2= 8 Note: |−6 +2| =|− 6|+|2| ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 Chapter 1 — Functions and relations 21 Consider two pointsAand Bon a number line: OAB ba On a number line the distance between points Aand Bis |a − b|=| b− a|. Thus | x − 2|≤3 can be read as ‘on the number line, the distance of xfrom 2 is less than or equal to 3’, and |x |≤3 can be read as ‘on the number line, the distance of xfrom the origin is less than or equal to 3’. Note that |x |≤3 is equivalent to −3≤x≤ 3or x∈ [−3, 3]. The graph of the function f: R → R, f (x ) =| x| is as shown here. x (–1, 1) 0(1, 1) y Note that |x |=|− x|, i.e. |x | is an even function. Example 11 Illustrate each of the following sets on a number line and represent the sets using interval notation. a {x :|x | < 4} b{x :|x |≥4} c{x :|x − 1|≤4} Solution a (−4, 4) 10–1–2–3 –4 2 3 4 b (−∞, −4]∪[4, ∞) 0–2 –4 2 4 c [−3, 5] 10–1–2–3 2 3 4 5 Example 12 Sketch the graphs of each of the following functions and state the range of each of the functions: af(x ) =| x− 3|+1 bf(x ) =−| x− 3|+1 ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

22 Essential Mathematical Methods3&4CAS Solution First, note that |a − b|= a− bif a ≥ band |a − b|=b −aif b≥ a. a f (x ) =| x− 3|+1 = x − 3+ 1if x≥ 3 3 − x+ 1if x< 3 = x − 2if x≥ 3 4 − x ifx< 3 x (3, 1) (0, 4) y 0 Range =[1, ∞) b f (x ) =−| x− 3|+1 = −( x− 3) +1if x≥ 3 −(3 −x) + 1if x< 3 = − x+ 4if x≥ 3 −2 +x ifx< 3 x 02 4 (3, 1) (0, –2) y Range =(−∞, 1] Using the TI-Nspire Complete as follows: Define f( x ) = abs ( x − 3) + 1 The absolute value function can be obtained by typing abs, found as a command in the catalog (k1A ) or found as a template using /+b>Math Templates. This can also be found by using the template key r. Open aGraphs application (/+I>Graphs) and let f1( x ) = f( x ) . Press enter to obtain the graph. Note that the expression abs( x − 3) + 1 could have been entered directly for f1( x ) . ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans, Kay Lipson, Peter Jones, Sue Avery 2012 Cambridge University Press

Chapter 1 — Functions and relations 23 Using the Casio ClassPad Ta pInteractive-Define and enter the function name, variable and function as shown. To enter the absolute value, press k and look in the to find the symbol. In enter f( x ) into y1, tick the box to select and tap to create the graph. Note that the expression could be directly entered in the y 1 = line but this gives you greater flexibility to use the function in other ways if required. Exercise 1D 1Evaluate each of the following: a |−5|+3 b|−5|+|− 3| c|−5|−|− 3| d |−5|−|− 3|−4 e|−5|−|− 3|−|−4| f|−5|+|− 3|−|−4| 2 On a number line, illustrate each of the following sets and represent the sets using interval notation: a {x :|x | < 3} b{x :|x |≥5} c{x :|x − 2|≤1} d {x :|x − 2|

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 24 Essential Mathematical Methods3&4CAS 1.5 Sums and products of functions The domain offis denoted by dom fand the domain of gby dom g. Letfand gbe functions such that dom f∩ dom g=∅ . The sum, f+ g, and the product, fg, as functions on dom f∩ dom gare defined by: 1 (f + g)( x) = f(x ) + g(x ) and 2(fg )(x) = f(x )g (x ) The domain of both f+ gand fgis the intersection of the domains of fand g, i.e. the values of xfor which both fand gare defined. Graphing sums of functions will be discussed in Section 3.9. Example 13 If f(x ) = \b x − 2 for all x≥ 2 and g(x ) = \b 4 − xfor all x≤ 4, find: a f+ g b(f + g)(3) cfg d(fg)(3) Solution a dom f∩ dom g= [2, 4] ( f + g)( x) = f(x ) + g(x ) = \b x − 2+ \b 4 − x dom ( f+ g) = [2, 4] b (f + g)(3) =\b 3 − 2+ \b 4 − 3 = 2 c (fg )(x) = f(x )g (x ) = ( x − 2)(4 −x) dom ( fg)= [2, 4] d (fg )(3) = (3 −2)(4 −3) = 1 Exercise 1E 1For each of the following, find ( f+ g)( x) and ( fg)(x) and state the domain for both f+ g and fg: a f(x ) = 3x ,g (x ) = x+ 2 b f(x ) = 1− x 2for all x∈ [−2, 2] and g(x ) = x 2for all x∈ R + c f(x ) = \b x and g(x ) = 1 \bxfor x∈ [1, ∞) d f(x ) = x 2,x ≥ 0 and g(x ) = \b 4 − x,0 ≤ x≤ 4 2 Functions f, g ,h , and kare defined by: i f(x ) = x 2+1, x∈ R iig(x ) = x,x ∈ R iii h(x ) = 1 x2, x = 0 iv k(x ) = 1 x, x = 0 a State which of the above functions are odd and which are even. b Form the functions of f+ h, fh ,g + k, gk ,f+ g, fg , stating which are odd and which are even. ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 Chapter 1 — Functions and relations 25 1.6 Composite functions A function may be considered to be similar to a machine for which the input (domain) is processed to produce an output (range).For example, the following diagram represents an ‘f-machine’ where f(x ) = 3x + 2 INPUT 11 3 OUTPUT f(3) = 3 × 3 + 2 = 11 f- machine An alternative diagram is: Domain, R 31 1 Range, R f With many processes, more than one machine operation is required to produce an output. Suppose an output is the result of one function being applied after another, e.g., f(x ) = 3x + 2 followed by g(x ) = x 2 This is illustrated diagrammatically on the right. INPUT OUTPUT 11 3 121 f(3) = 3 × 3 + 2 = 11g(11) = 11 2 = 121 f- machine g- machine A new function his formed. The rule for his h(x ) = (3x+ 2) 2 The diagram shows f(3) =11 and then g(11) =121. This may be written: h(3) =g( f(3)) =g(11) =121 Similarly, h(−2) =g( f(−2)) =g(−4) =16 h is said to be the compositionofgwith f. This is written h= g◦f(read ‘composition of ffollowed by g’) and the rule for his defined by h(x ) = g( f(x )). In the example we have considered: h (x ) = g( f(x )) = g(3 x+ 2) = (3x+ 2) 2 The domain of the function h= g◦ f= domain of f. In general for the composition of gwith fto be defined, range of f⊆ domain of g. When this composition (or composite function) of gwith fis defined it is denoted g◦f. For functions fand gwith domains Xand Yrespectively and such that the range of f⊆ Y,we define the composite function of gwith f: g ◦ f: X → R, where g ◦ f(x ) = g( f(x )) x X = domain of f range of f Y = domain of g fg g( f(x)) g( f(x)) f(x) ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

26 Essential Mathematical Methods3&4CAS Example 14 Find both f◦ g and g◦ f, stating the domain and range of each where: f : R → R, f(x ) = 2x − 1 and g:R → R,g (x ) = 3x 2 Solu tion To determine the existence of a composite function, it is useful to form a table of domains and ranges. Domain Range g R R+∪{0} f R R f◦ g is defined since ran g⊆ dom f, and g◦fis defined since ran f⊆ dom g. f ◦ g(x ) = f(g (x )) = f(3 x 2) = 2(3 x 2)− 1 = 6x 2− 1 and dom f◦ g = dom g= Rand ran f◦ g = [−1, ∞) g ◦ f(x ) = g( f(x )) = g(2 x− 1) = 3(2 x− 1) 2 = 12x 2−12 x+ 3 dom g◦ f= dom f = R ran g◦ f= [0, ∞) It can be seen from this example that in general f◦ g = g◦ f. Using the TI-Nspire Define f( x ) = 2x − 1 and g( x ) = 3x 2. The rules for f◦ g and g◦ fcan now be found using f( g ( x )) and g( f( x )) . ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans, Kay Lipson, Peter Jones, Sue Avery 2012 Cambridge University Press

Chapter 1 — Functions and relations 27 Using the Casio ClassPad Definef(x ) = 2x − 1 and g(x ) = 3x 2. The rules for f◦ g and g◦ fcan now be found using f( g ( x )) and g( f( x )) . Exa mple 15 For the functions g(x ) = 2x − 1, x∈ Rand f(x ) = √ x ,x ≥ 0: a State which of f◦ g and g◦ fis defined. b For the composite function that is defined, state the domain and rule. Solu tion aRange of f⊆ domain of g but range of g  domain off. ∴ g ◦ fis defined but f◦ g is not defined. Domain Range g R R f R+∪{ 0} R+∪{0} b g◦ f(x ) = g( f(x )) = g(√ x ) = 2√ x − 1 dom g◦ f= dom f= R +∪{ 0} Exa mple 16 For the functions f(x ) = x 2−1, x∈ R, and g(x ) = √ x ,x ≥ 0: a State why g◦ fis not defined. b Define a restriction f ∗offsuch that g◦ f ∗is defined and find g◦ f ∗. ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans, Kay Lipson, Peter Jones, Sue Avery 2012 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 28 Essential Mathematical Methods3&4CAS Solution aRange of f  domain of g. ∴ g ◦ fis not defined. Domain Range f R [−1, ∞) g R+∪{0} R+∪{0} b For g◦ fto be defined, range of f⊆ domain of g, i.e. range of f ⊆ R +∪{ 0}. For range of fto be a subset of R +∪{0}, the domain of f must be restricted to a subset of: {x :x ≤− 1}∪{ x:x ≥ 1}, orR\(−1, 1). So we define f ∗by: y= f(x) x 0–1 –1 1 y f∗: R \(−1, 1)→ R, f ∗(x ) = x 2−1 g ◦ f ∗(x ) = g( f ∗(x )) = g(x 2− 1) = x2− 1 dom g◦ f ∗= dom f ∗=R\(−1, 1) The composite function g◦ f ∗is: g ◦ f ∗:R \(−1, 1)→ R,g ◦ f ∗(x ) = x2− 1 Compositions involving the modulus function Functions with rules of the form y=| f(x )| and y= f(| x|) are considered in this section. Functions of the form y == ||| f(x)||| |f| is the composition g◦ fwhere g(x ) =| x|. The function fis applied first and then the modulus function. The following observation enables the graph of functions with rule of the form y=| f(x )| to be sketched if the graph of y= f(x ) is known: | f(x )|= f(x )if f(x ) ≥ 0 and |f(x )|=− f(x )if f(x ) < 0 Example 17 Sketch the graphs of each of the following: a y=| x 2−4| by=| 2 x−1| ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 Chapter 1 — Functions and relations 29 Solution aThe graph of y= x 2−4 is drawn and the negative part reflected in the x-axis. x 0 –2 2 y b The graph of y= 2 x−1is drawn and the negative part reflected in the x-axis. x y = –1 y = 1 0 y Functions of the form y == f(|||x|||) The graphs of functions with rules of the form y= f(| x|) where x∈ Rare sketched by reflecting the graph of y= f(x ), for x≥ 0, in the y-axis. The function with rule f(| x|)isthe result of the composition f◦ g where g(x ) =| x|. Example 18 Sketch the graphs of each of the following: a y=| x| 2− 2|x| (This is the rule for the function f◦ g where f(x ) = x 2−2x and g (x ) =| x|.) b y= 2 |x | (This is the rule for the function f◦ g where f(x ) = 2 xand g(x ) =| x|.) Solution a x –2 2 0 y The graph of y= x 2−2x ,x ≥ 0, is reflected in the y-axis. b x 1 0 y The graph of y= 2 x,x ≥ 0, is reflected in the y-axis. ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 30 Essential Mathematical Methods3&4CAS Exercise1F 1For each of the following, find f(g (x )) and g( f(x )): a f(x ) = 2x − 1, g(x ) = 2x bf(x ) = 4x + 1, g(x ) = 2x + 1 c f(x ) = 2x − 1, g(x ) = 2x − 3 df(x ) = 2x − 1, g(x ) = x 2 e f(x ) = 2x 2+ 1, g(x ) = x− 5 ff(x ) = 2x + 1, g(x ) =| x| 2 For the functions f(x ) = 2x − 1 and h(x ) = 3x + 2, find: a f◦ h(x ) bh(f(x )) cf◦ h(2) dh◦ f(2) e f(h (3)) fh(f(−1)) gf◦ h(0) 3 For the functions f(x ) = x 2+2x and h(x ) = 3x + 1, find: a f◦ h(x ) bh◦ f(x ) cf◦ h(3) d h◦ f(3) ef◦ h(0) fh◦ f(0) 4 For the functions h:R \{0}→ R,h (x ) = 1 x2and g:R +→ R,g (x ) = 3x + 2, find: a h◦g (state rule and domain) bg◦h (state rule and domain) c h◦g(1) dg◦h(1) 5 fand gare the functions given by f: R → R, f(x ) = x 2−4 and g :R +∪{ 0}→ R,g (x ) = \b x . a State the ranges of fand g. bFind f◦ g, stating its range. c Explain why g◦ fdoes not exist. 6 Let fand gbe functions given by: f: R \{0}→ R, f(x ) = 1 2 1 x + 1 g :R \ 1 2 → R,g (x ) = 1 2 x − 1 Find: a f◦ g bg◦ f, and state the range in each case 7 The functions fand gare defined by f: R → R, f(x ) = x 2−2 and g :{x :x ≥ 0}→ R, where g(x ) = \b x . a Explain why g◦ fdoes not exist. bFind f◦ g and sketch its graph. 8a Forf(x ) = 4− xand g(x ) =| x|, find f◦ g and g◦ fand sketch the graphs of each of these functions. b For f(x ) = 9− x 2and g(x ) =| x|, find f◦ g and g◦ fand sketch the graphs of each of these functions. c For f: R \{0) →R, f(x ) = 1 xand g:R \{0) →R,g (x ) =| x|, find f◦ g and g◦ f and sketch the graphs of each of these functions. ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 Chapter 1 — Functions and relations 31 9f:{x :x ≤ 3}→ R, f(x ) = 3− xand g:R → R,g (x ) = x 2−1 a Show that f◦ g is not defined. b Define a restriction g *ofgsuch that f◦ g *is defined and find f◦ g *. 10 f: R +→ R, f(x ) = x −12and g:R → R,g (x ) = 3− x a Show that f◦ g is not defined. b By suitably restricting the domain of g, obtain a functiong 1such thatf◦ g 1is defined. 11 Letf: R → R, f(x ) = x 2and let g:{x :x ≤ 3}→ R,g (x ) = \b 3 − x.State with reasons whether: a f◦ g exists bg◦ fexists 12 Letf: S → R, f(x ) = \b 4 − x 2and Sbe the set of all real values of xfor which f(x )is defined. Let g:R → R, where g(x ) = x 2+1. a Find S. bFind the range of fand the range of g. c State whether or not f◦ g and g◦ fare defined and give a reason for each assertion. 13 Letabe a positive number, let f:[2 ,∞) →R, f(x ) = a− xand let g:( −∞, 1]→ R, g (x ) = x 2+a. Find all values of afor which f◦ g and g◦ fboth exist. 1.7 Inverse functions If fis a one-to-one function, then for each number yin the range of fthere is exactly one number, x, in the domain of fsuch that f(x ) = y. Thus if fis a one-to-one function, a new function f −1, called the inverse off,maybe defined by: f −1(x ) = yif f( y ) = x, for x∈ ran f, y ∈ dom f It is not difficult to see what the relation between fand f −1 means geometrically. The point (x, y)isonthe graph of f −1 if the point ( y,x) is on the graph of f. Therefore, to get the graph of f −1 from the graph of f, the graph of fis to be reflected in the line y= x. (y, x) (x, y)y = x f –1 f x 0y From this the following is evident: domf −1 =ran f ran f −1 =dom f A function has an inverse function if and only if it is one-to-one. We note f◦ f −1(x ) = x, for all x∈ dom f −1 f−1 ◦f(x ) = x, for all x∈ dom f ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 32 Essential Mathematical Methods3&4CAS Example 19 Find the inverse functionf −1 of the functionf(x ) = 2x − 3. Solution Method 1 The graph of fhas equation y= 2x − 3 and the graph of f −1 has equation x = 2y − 3, i.e. xand yare interchanged. Solve for y. x + 3= 2y and y= 1 2( x + 3) ∴ f−1(x ) = 1 2( x + 3) and dom f −1 =ran f = R Method 2 We require f −1 such that: f ( f −1(x )) = x ∴ 2f −1(x ) − 3= x ∴ f−1(x ) = 1 2( x + 3) and dom f −1 =ran f = R Example 20 fis the function defined by f(x ) = 1 x2, x ∈ R\{0}. Define a suitable restriction for f, f ∗, such that f ∗−1 exists. Solution fis not a one-to-one function. Therefore the inverse function f −1 is not defined. The following restricted functions of fare one-to-one. f 1:(0 ,∞) →R, f 1(x ) = 1 x2 Range f 1=(0, ∞) f 2:( −∞, 0)→ R, f 2(x ) = 1 x2 Range f 2=(0, ∞) Let f ∗bef 1and determine f 1−1 . x 0 y ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 Chapter 1 — Functions and relations 33 Method 1 Interchangingxand y: 01 1 x f1(x) = 1 x2 y = x f 1–1 (x) = 1 √x y x = 1 y2 y2=1 x ∴ y =± 1 \bx But range f −1 1 = domain f 1 =(0, ∞) ∴ f−1 1 = 1 \bx, ran f −1 1 = (0, ∞) and dom f −1 1 = ran f 1=(0, ∞) ∴ f−1 1:(0 ,∞) →R, f −1 1( x ) = 1 \bx Method 2 We require f −1 1 such that: f 1 f −1 1( x ) = x ∴ 1  f −1 1( x ) 2= x ∴ f−1 1( x ) =± 1 \bx But range f −1 1 = domain f 1 =(0, ∞) ∴ f−1 1 = 1 \bx, ran f −1 1 = (0, ∞) and dom f −1 1 = ran f 1=(0, ∞) ∴ f−1 1:(0 ,∞) →R, f −1 1( x ) = 1 \bx Exercise 1G 1For each of the following, find the rule for the inverse: a f: R → R, f(x ) = x− 4 bf: R → R, f(x ) = 2x c f: R → R, f(x ) = 3 x 4 d f: R → R, f(x ) = 3 x − 2 4 2 Find the inverse of each of the following functions, stating the domain and range for each: a f:[ −2, 6]→ R, f(x ) = 2x − 4 bg(x ) = 1 9 − x, x > 9 c h(x ) = x 2+2, x≥ 0 df:[ −3, 6]→ R, f(x ) = 5x − 2 e g:(1 ,∞) →R,g (x ) = x 2−1 fh:R +→ R,h (x ) = \b x ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 34 Essential Mathematical Methods3&4CAS 3Consider the function g:[ −1, ∞)→R,g (x ) = x 2+2x . a Find g −1, stating the domain and range. bSketch the graph of g −1. 4 Let f: S → R, where S={ x:0 ≤x≤ 3} and f(x ) = 3− 2x . Find f −1(2) and the domain of f −1. 5 Find the inverse of each of the following functions, stating the domain and range of each: a f:[ −1, 3]→ R, f(x ) = 2x bf:[0 ,∞) →R, f(x ) = 2x 2− 4 c {(1, 6),(2,4),(3,8),(5,11)} dh:R −→ R,h (x ) = \b − x e f: R → R, f(x ) = x 3+1 fg:( −1, 3)→ R,g (x ) = (x + 1) 2 g g:[1 ,∞) →R,g (x ) = \b x − 1 hh:[0 ,2] → R,h (x ) = \b 4 − x 2 6 For each of the following functions, sketch the graph of the function and on the same set of axes sketch the graph of the inverse function. For each of the functions state the rule, domain and range of the inverse. It is advisable to draw in the line with equation y= xfor each set of axes. a y= 2x + 4 b f(x ) = 3 − x 2 c f:[2 ,∞) →R, f(x ) = (x − 2) 2 d f:[1 ,∞) →R, f(x ) = (x − 1) 2 e f:( −∞, 2]→ R, f(x ) = (x − 2) 2 f f: R +→ R, f(x ) = 1 x g f: R +→ R, f(x ) = 1 x2 h h(x ) = 1 2( x − 4) 7 Copy each of the following graphs and on the same set of axes draw the inverse of each of the corresponding functions: x (3, 3) (0, 0) y ab x (2, 1) (3, 4) 01 y c x 2 03 y d x 01 –4 y ef x –3 3 0 y x y (1, 1) (0, 0) (–1, –1) ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

P1: FXS/ABE P2: FXS 9780521740531c01.xml CUAU156-EVANS August 8, 2011 12:10 Chapter 1 — Functions and relations 35 gh x 0 y x 0 –2 y 8 Match each of the graphs of a,b, cand dwith its inverse. ab x 0 y x 0 y cd x 0–1 y x = – 1 x 0 y = 1 y AB x x = 1 0 y x 0 y CD x 0 y y = –1 x 0 y ISBN 978-1-107-67685-5 Photocopying is restricted under law and this material must not be transferred to another party. � Michael Evans et al. 2011 Cambridge University Press

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