Giáo trình toán tiếng anh dành cho học sinh theo học chương trình tú tài quốc tế hệ higher level

position vector positive definite positively skewed principal axis probability density function probability distribution product rule proposition quadratic formula quadratic function qua[r]

for the international student

Mathematics

Specialists in mathematics publishing

Mathematics HL (Core)

Paul Urban David Martin Robert Haese Sandra Haese

Mark Humphries Michael Haese

for use with

IB Diploma Programme

second edition

Mathematics HL (Core) second edition

This book is copyright

Copying for educational purposes

Acknowledgements

Disclaimer

Paul Urban B.Sc.(Hons.),B.Ec

David Martin B.A.,B.Sc.,M.A.,M.Ed.Admin

Robert Haese B.Sc

Sandra Haese B.Sc

Mark Humphries B.Sc.(Hons.)

Haese & Harris Publications

3 Frank Collopy Court, Adelaide Airport, SA 5950, AUSTRALIA

Telephone: +61 8 8355 9444, Fax: + 61 8 8355 9471

Email:

National Library of Australia Card Number & ISBN 978-1-876543-11-2

© Haese & Harris Publications 2008

Published by Raksar Nominees Pty Ltd

3 Frank Collopy Court, Adelaide Airport, SA 5950, AUSTRALIA

First Edition 2004

2005 three times , 2006, 2007Second Edition 2008

Cartoon artwork by John Martin Artwork by Piotr Poturaj and David Purton

Cover design by Piotr Poturaj

Computer software by David Purton, Thomas Jansson and Troy Cruickshank

Typeset in Australia by Susan Haese (Raksar Nominees) Typeset in Times Roman 10 /11

The textbook and its accompanying CD have been developed independently of the InternationalBaccalaureate Organization (IBO) The textbook and CD are in no way connected with, orendorsed by, the IBO

Except as permitted by the Copyright Act (any fair dealing for thepurposes of private study, research, criticism or review), no part of this publication may bereproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic,mechanical, photocopying, recording or otherwise, without the prior permission of the publisher.Enquiries to be made to Haese & Harris Publications

: Where copies of part or the whole of the book are madeunder Part VB of the Copyright Act, the law requires that the educational institution or the bodythat administers it has given a remuneration notice to Copyright Agency Limited (CAL) Forinformation, contact the Copyright Agency Limited

: While every attempt has been made to trace and acknowledge copyright, theauthors and publishers apologise for any accidental infringement where copyright has proveduntraceable They would be pleased to come to a suitable agreement with the rightful owner

: All the internet addresses (URL’s) given in this book were valid at the time ofprinting While the authors and publisher regret any inconvenience that changes of address maycause readers, no responsibility for any such changes can be accepted by either the authors or thepublisher

Michael Haese B.Sc.(Hons.),Ph.D

Reprinted (with minor corrections)

Reprinted (with minor corrections)

\Qw_ \Qw_\\"

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Web:

Mathematics for the International Student: Mathematics HL has been written to reflect the

syllabus for the two-year IB Diploma Mathematics HL course It is not our intention to define thecourse Teachers are encouraged to use other resources We have developed the book independently

of the International Baccalaureate Organization (IBO) in consultation with many experiencedteachers of IB Mathematics The text is not endorsed by the IBO

This second edition builds on the strengths of the first edition Many excellent suggestions werereceived from teachers around the world and these are reflected in the changes In some casessections have been consolidated to allow for greater efficiency Changes have also been made inresponse to the introduction of a calculator-free examination paper A large number of questions,including some to challenge even the best students, have been added In particular, the final chaptercontains over 200 miscellaneous questions, some of which require the use of a graphics calculator.These questions have been included to provide more difficult challenges for students and to givethem experience at working with problems that may or may not require the use of a graphicscalculator

The combination of textbook and interactive Student CD will foster the mathematical development

of students in a stimulating way Frequent use of the interactive features on the CD is certain tonurture a much deeper understanding and appreciation of mathematical concepts

The book contains many problems from the basic to the advanced, to cater for a wide range ofstudent abilities and interests While some of the exercises are simply designed to build skills,every effort has been made to contextualise problems, so that students can see everyday uses andpractical applications of the mathematics they are studying, and appreciate the universality ofmathematics

Emphasis is placed on the gradual development of concepts with appropriate worked examples, but

we have also provided extension material for those who wish to go beyond the scope of thesyllabus Some proofs have been included for completeness and interest although they will not beexamined

For students who may not have a good understanding of the necessary background knowledge forthis course, we have provided printable pages of information, examples, exercises and answers onthe Student CD To access these pages, simply click on the ‘Background knowledge’ icons whenrunning the CD

It is not our intention that each chapter be worked through in full Time constraints will not allowfor this Teachers must select exercises carefully, according to the abilities and prior knowledge oftheir students, to make the most efficient use of time and give as thorough coverage of work aspossible

Investigations throughout the book will add to the discovery aspect of the course and enhancestudent understanding and learning Many Investigations could be developed into portfolioassignments Teachers should follow the guidelines for portfolio assignments to ensure they setacceptable portfolio pieces for their students that meet the requirement criteria for the portfolios.Review sets appear at the end of each chapter and a suggested order for teaching the two-yearcourse is given at the end of this Foreword

The extensive use of graphics calculators and computer packages throughout the book enablesstudents to realise the importance, application and appropriate use of technology No single aspect

of technology has been favoured It is as important that students work with a pen and paper as it isthat they use their calculator or graphics calculator, or use a spreadsheet or graphing package oncomputer

The interactive features of the CD allow immediate access to our own specially designed geometrypackages, graphing packages and more Teachers are provided with a quick and easy way todemonstrate concepts, and students can discover for themselves and re-visit when necessary

dents These instructions are written for Texas Instruments and Casio calculators.

In this changing world of mathematics education, we believe that the contextual approach shown in thisbook, with the associated use of technology, will enhance the students’ understanding, knowledge andappreciation of mathematics, and its universal application

We welcome your feedback

Particular thanks go to Stephen Hobbs who has given generously of his time in reviewing the firstedition and making suggestions for improvement in this second edition Thanks are also due to

Dr Andrzej Cichy, Peter Blythe, Brendan Watson, Myrricia Holmann, Jeff Jones, Mark Willis,John Poole and Marjut Mäenpää We acknowledge the contributions of John Owen and Mark Bruce inthe preparation of the first edition and we also want to thank others who provided assistance – theyinclude: Cameron Hall, Fran O'Connor, Glenn Smith, Anne Walker, Malcolm Coad, Ian Hilditch,Phil Moore, Julie Wilson, Kerrie Clements, Margie Karbassioun, Brian Johnson, Carolyn Farr,Rupert de Smidt, Terry Swain, Marie-Therese Filippi, Nigel Wheeler, Sarah Locke, Rema George.The publishers wish to make it clear that acknowledging these individual does not imply anyendorsement of this book by any of them and all responsibility for content rests with the authors andpublishers

Teachers are encouraged to carefully check the BACKGROUND KNOWLEDGE sections supplied

on the accompanying CD to ensure that basics have been mastered relatively early in the two-year

HL course Some of these topics naturally occur at the beginning of a specific chapter, as indicated

in the table of contents Click on the BACKGROUND KNOWLEDGE active icons to access theprintable pages on the CD

Teachers will have their personal preferences for the order in which the chapters are tackled Asuggestion is to work progressively from Chapter 1 through to Chapter 20, but leave Chapters 9, 15and, possibly, 16 for the second year The remaining chapters can be worked through in order

Alternatively, for the first year, students could work progressively from Chapter 1 to Chapter 23 butnot necessarily including chapters 7, 15 and 16 Chapter 9 ‘Mathematical Induction’ could also beattempted later, perhaps early in the second year In some parts of the world, the topics ofPolynomials, Complex Numbers, 3-D Vector Geometry and Calculus are not usually covered untilthe final year of school

Another approach could be to teach just those topics that are included in the Mathematics SLsyllabus in the first year and leave the remaining topics for completion in the second year

However, it is acknowledged that there is no single best way for all teachers to work through thesyllabus Individual teachers have to consider particular needs of their students and otherrequirements and preferences that they may have

TEACHING THE TWO-YEAR COURSE – A SUGGESTED ORDER

NOTE ON ACCURACY

HL & SL COMBINED CLASSES

HL OPTIONS

SUPPLEMENTARY BOOKS

The CD is ideal for independent study Frequent use will nurture a deeper

understanding of Mathematics Students can revisit concepts taught in class and

undertake their own revision and practice The CD also has the text of the book,

allowing students to leave the textbook at school and keep the CD at home

The icon denotes an Interactive Link on the CD Simply ‘click’ the icon to access

a range of interactive features:

For those who want to make sure they have the prerequisite levels of understanding for this course,printable pages of background information, examples, exercises and answers are provided on the CD.Click the ‘Background knowledge’ icon on pages 12 and 248

Graphics calculators: Instructions for using graphics calculators are also given on the CD and can beprinted Instructions are given for Texas Instruments and Casio calculators Click on the relevant icon(TI or C) to access printable instructions

Examples in the textbook are not always given for both types of calculator Where

that occurs, click on the relevant icon to access the instructions for the other type

This is a companion to the textbook It offers coverage

of each of the following options:

In addition, coverage of the Geometry option for students undertaking the

IB Diploma course is presented on the CD that accompanies the

graphing and geometry software

graphics calculator instructions

computer demonstrations and simulations

background knowledge (as printable pages)

Topic 8 –Statistics and probability

Topic 9 – Sets, relations and groups

Topic 10 – Series and differential equations

Topic 11 – Discrete mathematics

TIC

SYMBOLS AND NOTATION

USED IN THIS BOOK 10

to access, ‘click’ active icon on CD

B Scientific notation (Standard form) CD

C Number systems and set notation CD

E Linear equations and inequalities CD

F Modulus or absolute value CD

J Adding and subtracting algebraic fractions CD

K Congruence and similarity CD

B Function notation, domain and range 21

E Inequalities (inequations) 32

H Asymptotes of other rational functions 42

F Graphs of exponential functions 88

H The natural exponential ‘ ’ 95

E Exponential equations using logarithms 112

G Graphs of logarithmic functions 115

C Simple rational functions 133

D Further graphical transformations 137

A Solving quadratic equations (Review) 145

B The discriminant of a quadratic 149

C The sum and product of the roots 152

D Graphing quadratic functions 153

E Finding a quadratic from its graph 161

G Problem solving with quadratics 167

A The process of induction 234

B The principle of mathematical induction 236

C Indirect proof (extension) 244

BACKGROUND KNOWLEDGE –

TRIGONOMETRY WITH RIGHT

ANGLED TRIANGLES – Printable pages CD

B Arc length and sector area 250

C The unit circle and the basic

C Using the sine and cosine rules 277

A Observing periodic behaviour 285

C Modelling using sine functions 293

G Using trigonometric models 305

H Reciprocal trigonometric functions 307

I Trigonometric relationships 309

L Trigonometric equations in quadratic form 318

M Trigonometric series and products 318

B Matrix operations and definitions 326

C The inverse of a 2 × 2 matrix 342

D 3 × 3 and larger matrices 348

E Solving systems of linear equations 350

F Solving systems using row operations 354

C 2-D vectors in component form 383

E 3-D vectors in component form 390

F Algebraic operations with vectors 393

I The scalar product of two vectors 402

J The vector product of two vectors 407

A Complex numbers as 2-D vectors 422

B Modulus, argument, polar form 425

D Roots of complex numbers 441

E Further complex number problems 445

B Applications of a line in a plane 456

8 COUNTING AND THE

C Relationship between lines 461

F The intersection of two or more planes 473

A Continuous numerical data and histograms 485

B Measuring the centre of data 489

D Measuring the spread of data 502

E Statistics using technology 510

F Variance and standard deviation 512

G The significance of standard deviation 518

F Sampling with and without replacement 543

B Derivatives at a given -value 595

C Simple rules of differentiation 600

E Product and quotient rules 607

C Motion in a straight line 627

E Some special exponential functions 683

A Derivatives of circular functions 688

B The derivatives of reciprocal circular

22 DERIVATIVES OF EXPONENTIALAND LOGARITHMIC

23 DERIVATIVES OF CIRCULARFUNCTIONS AND RELATED

C Problem solving by integration 748

B Further integration by substitution 769

D Miscellaneous integration 773

E Separable differential equations 774

A Discrete random variables 786

B Discrete probability distributions 788

D The measures of a discrete random

E The binomial distribution 801

F The Poisson distribution 807

D Applications of the normal distribution 828

a 2

¡ x2

N the set of positive integers and zero,

f0, 1, 2, 3, g

Z the set of integers, f0, §1, §2, §3, g

Z + the set of positive integers, f1, 2, 3, g

Q the set of rational numbers

Q + the set of positive rational numbers,

fx j x > 0 , x 2 Q g

R the set of real numbers

R + the set of positive real numbers,

arg z the argument of z

Rez the real part of z

Imz the imaginary part of z

fx 1 , x2, g the set with elements x1, x2,

n(A) the number of elements in the finite set A

fx j the set of all x such that

2 is an element of

=

2 is not an element of

? the empty (null) set

U the universal set

· or 6 is less than or equal to

is not greater than

is not less than [ a , b ] the closed interval a 6 x 6 b ] a, b [ the open interval a < x < b

u n the nth term of a sequence or series

d the common difference of an arithmetic

sequence

r the common ratio of a geometric sequence

S n the sum of the first n terms of a sequence,

u 1 + u 2 + ::::: + u n

S1 or S the sum to infinity of a sequence,

u 1 + u 2 + :::::

nXi=1

f : A ! B f is a function under which each element of

set A has an image in set B

f : x 7! y f is a function under which x is mapped to y

f (x) the image of x under the function f

f ¡1 the inverse function of the function f

f ± g the composite function of f and g lim

x !a f (x) the limit of f (x) as x tends to a dy

dx the derivative ofy with respect to x

f 0 (x) the derivative of f (x) with respect to x

d 2 y

dx 2 the second derivative of y with respect to x

f 00 (x) the second derivative of f (x) with respect to x

d n y

dx n the nth derivative of y with respect to x

f (n) (x) the nth deriviative of f (x) with respect to x

R

y dx the indefinite integral of y with respect to x

Z b a

y dx the definite integral of y with respect to x between the limits x = a and x = b

e x exponential function of x logax logarithm to the base a of x

sin, cos, tan the circular functions

arcsin,

arccos,

arctan

¾

the inverse circular functions

csc, sec, cot the reciprocal circular functions

A(x, y) the point A in the plane with Cartesian

coordinates x and y

[AB] the line segment with end points A and B

AB the length of [AB]

(AB) the line containing points A and B

b

A the angle at A

[

CAB or CbAB the angle between [CA] and [AB]

¢ABC the triangle whose vertices are A, B and C

v the vector v

¡!

AB the vector represented in magnitude and

direction by the directed line segment from

A to B

a the position vector ¡!OA

i, j, k unit vectors in the directions of the Cartesian

coordinate axes

j a j the magnitude of vector a

j¡!AB j the magnitude of ¡!AB

v ² w the scalar product of v and w

v £ w the vector product of v and w

A¡1 the inverse of the non-singular matrix A

detA or jAj the determinant of the square matrix A

I the identity matrix

P 0 (A) probability of the event “not A”

P (A j B) probability of the event A given B

x1, x2, observations of a variable

f1, f2, frequencies with which the observations

x1, x2, x3, occur

p x probability distribution function P(X = x)

of the discrete random variable X

f (x) probability density function of the continuous

random variable X E(X) the expected value of the random variable X Var (X) the variance of the random variable X

N(¹, ¾ 2 ) normal distribution with mean ¹ and variance ¾ 2

X » B(n, p) the random variable X has a binomial

distribution with parameters n and p

X » Po(m) the random variable X has a Poisson

distribution with mean m

X » N(¹, ¾ 2 ) the random variable X has a normal

distribution with mean ¹ and variance ¾ 2

SUMMARY OF CIRCLE PROPERTIES

BACKGROUND KNOWLEDGE

² A circle is a set of points which are equidistant from

a fixed point, which is called its centre.

² The circumference is the distance around the entire

circle boundary

² An arc of a circle is any continuous part of the circle.

² A chord of a circle is a line segment joining any two

points of a circle

arc chord

centre

circle

² A semi-circle is a half of a circle.

² A diameter of a circle is any chord passing

² A radius of a circle is any line segment

² A tangent to a circle is any line which

diameter

radius

tangent point of contact

through its centre

joining its centre to any point on the circle

touches the circle in exactly one point

Before starting this course you can make sure that you have a good

understanding of the necessary background knowledge Click on

the icon alongside to obtain a printable set of exercises and

answers on this background knowledge

BACKGROUND KNOWLEDGE

Click on the icon to access printable facts about number sets

NUMBER SETS

AM= BMA

M B O

Click on the appropriate icon to revisit these well known theorems

The angle in asemi-circle is a rightangle

The perpendicularfrom the centre of acircle to a chordbisects the chord

The tangent to acircle is perpendicular

to the radius at thepoint of contact

Tangents from anexternal point areequal in length

The angle at thecentre of a circle istwice the angle on thecircle subtended bythe same arc

at the point of contact

is equal to the anglesubtended by thechord in the alternatesegment

GEOMETRY PACKAGE

GEOMETRY PACKAGE

GEOMETRY PACKAGE

GEOMETRY PACKAGE

GEOMETRY PACKAGE

P = 4 l P= 2(l + w) P = a + b + c l=( )µ

360 2¼ror

Shape Figure Formula

a

b

)

For some shapes we can derive formula for perimeter The

formulae for the most common shapes are given below:

The length of an arc

is a fraction of the circumference of a circle.

SUMMARY OF MEASUREMENT FACTS

PERIMETER FORMULAE

The distance around a closed figure is its perimeter.

VOLUME FORMULAE

Object

Volume of uniform solid

= area of end £ length

Relations and functionsFunction notation, domain and rangeComposite functions,

Sign diagramsInequalities (inequations)The modulus functionThe reciprocal functionAsymptotes of other rational functionsInverse functions

Functions which have inverses

Review set 1AReview set 1BReview set 1C

f±g

x

Car park charges

In mathematical terms, because we have a

relationship between two variables, time and

cost, the schedule of charges is an example

of a relation.

A relation may consist of a finite number of

ordered pairs, such as f(1, 5), (¡2, 3), (4, 3),

(1, 6)g or an infinite number of ordered pairs

The parking charges example is clearly the

lat-ter as any real value of time (t hours) in the

The set which describes the possibley-values is called the range of the relation.

For example: ² the range of the car park relation is f5, 9, 11, 13, 18, 22, 28g

² the range of f(1, 5), (¡2, 3), (4, 3), (1, 6)g is f3, 5, 6g

We will now look at relations and functions more formally

RELATIONS AND FUNCTIONS

A

The charges for parking a car in a short-term car park at an

airport are given in the table shown alongside

There is an obvious relationship between the time spent in

the car park and the cost The cost is dependent on the

length of time the car is parked

Looking at this table we might ask: How much would be

charged for exactly one hour? Would it be $5 or $9?

To make the situation clear, and to avoid confusion, we

could adjust the table and draw a graph We need to indicate

that2-3 hours really means a time over 2 hours up to and

including3 hours, i.e., 2 < t 6 3

charge ($)

time ( )t

10 20 30

exclusion inclusion

A relation is any set of points on the Cartesian plane.

A relation is often expressed in the form of an equation connecting the variablesx and y.For example y = x + 3 and x = y2 are the equations of two relations

These equations generate sets of ordered pairs

Their graphs are:

However, a relation may not be able to be defined by an equation Below are two exampleswhich show this:

A function, sometimes called a mapping, is a relation in which no

two different ordered pairs have the samex-coordinate (first member)

We can see from the above definition that a function is a special type of relation

Algebraic Test:

If a relation is given as an equation, and the substitution of any value

for x results in one and only one value of y, we have a function

For example: ² y = 3x ¡ 1 is a function, as for any value of x there is only one value of y

² x = y2 is not a function since if x = 4, say, then y = §2

If we draw all possible vertical lines on the graph of a relation, the relation:

² is a function if each line cuts the graph no more than once

² is not a function if at least one line cuts the graph more than once

> 0, > 0

These 13 points form a relation.

FUNCTIONS

TESTING FOR FUNCTIONS

Geometric Test or “Vertical Line Test”:

Which of the following relations are functions?

y

x

y

x y

x

y

x y

x

² If a graph contains a small open circle such as , this point is not included.

² If a graph contains a small filled-in circle such as , this point is included.

² If a graph contains an arrow head at an end such as then the graph continuesindefinitely in that general direction, or the shape may repeat as it has done previously

3 Will the graph of a straight line always be a function? Give evidence.

4 Give algebraic evidence to show that the relation x2+ y2= 9 is not a function

Function machines are sometimes used to illustrate how functions behave.

For example:

So, if 4 is fed into the machine,2(4) + 3 = 11 comes out

The above ‘machine’ has been programmed to perform a particular function

Iff is used to represent that particular function we can write:

f is the function that will convert x into 2x + 3

So, f would convert 2 into 2(2) + 3 = 7 and

¡4 into 2(¡4) + 3 = ¡5

This function can be written as:

Two other equivalent forms we use are: f(x) = 2x + 3 or y = 2x + 3

So, f(x) is the value of y for a given value of x, i.e., y = f (x)

Notice that for f (x) = 2x + 3, f(2) = 2(2) + 3 = 7 and f(¡4) = 2(¡4) + 3 = ¡5:Consequently, f (2) = 7 indicates that the point (2, 7)

lies on the graph of the function

Likewise f (¡4) = ¡5 indicates that the point

(¡4, ¡5) also lies on the graph

FUNCTION NOTATION, DOMAIN AND RANGE

B

x

2 + 3x

I double the input and then add 3

If f : x 7! 2x2¡ 3x, find the value of: a f(5) b f (¡4)

b find a value ofx where G(x) does not exist

c find G(x + 2) in simplest form

a find V (4) and state what V (4) means

b findt when V (t) = 5780 and explain what this represents

c find the original purchase price of the photocopier

8 On the same set of axes draw the graphs of three different functionsf (x) such thatf(2) = 1 and f (5) = 3:

9

10 GivenT (x) = ax2+ bx + c, find a, b and c if T (0) = ¡4, T (1) = ¡2 and T (2) = 6:

The domain of a relation is the set of permissible values thatx may have

The range of a relation is the set of permissible values thaty may have

For example:

So, the domain is fx j x > ¡1g or x 2 [ ¡1, 1 [:All values of y > ¡3 are permissible

So, the range is fy j y > ¡3g or y 2 [ ¡3, 1 [

So, the domain is fx j x is in R g or x 2 R

So, the domain is fx j x 6= 2g.Likewise, the range is fy j y 6= 1g

that x > 3 and we write this as

So, the range is fy j y 6 1g or y 2 ] ¡ 1, 1 ]

The domain and range of a relation are often described using interval notation.

If the value of a photocopiert years after purchase is given by V (t) = 9650¡860t euros:

² the domain is fx j x > 0g

or x 2 [ 0, 1 [

² the range is fy j y 6 100g

or y 2 ] ¡ 1, 100 ]:

Intervals have corresponding graphs For example:

fx j x > 3g or x 2 [ 3, 1 [ is read “the set of allx such that x is greater than

or equal to3” and hasnumber line graph

fx j x < 2g or x 2 ] ¡1, 2 [ has number line graph

fx j ¡2 < x 6 1g or x 2 ] ¡2, 1 ] has number line graph

fx j x 6 0 or x > 4g has number line graph

1 Write down the domain and range for each of the following functions:

( -4 -3 , )

Use a graphics calculator to help sketch graphs of the following functions Find thedomain and range of each

INVESTIGATION 1 FLUID FILLING FUNCTIONS

The depth-time graph for the

case of a cylinder would be as

shown alongside:

The question arises:

‘What changes in appearance

of the graph occur for different

2 Use the water filling demonstration to check your answers to question 1

3 Write a brief report on the connection between the shape of a vessel and the responding shape of its depth-time graph You may wish to discuss this in parts.For example, first examine cylindrical containers, then conical, then other shapes.Gradients of curves must be included in your report

cor-4 Draw possible containers as in question 1 which have the following ‘depth v time’graphs:

When water is added at a to a cylindrical container, the depth

of water in the container is a linear function of time This is because thevolume of water added is directly proportional to the time taken to add it

If water was not added at a constant rate the direct proportionality wouldnot exist

constant rate

¡

¡

The ability to break down functions into composite functions is useful in differential calculus.

Given f : x 7! 2x + 1 and g : x 7! 3 ¡ 4x find in simplest form:

C

I double and then add 3

I raise the number to the power 4

Given f : x 7! f(x) and g : x 7! g(x), then the composite function of f and g

will convertx into f(g(x))

f ± g is used to represent the composite function of f and g

f ± g means “f following g” and (f ± g)(x) = f(g(x)) i.e., f ± g : x 7! f(g(x))

Notice how is following

f g

Note: If F (x) = (f ± g)(x), the domain of F is the domain of g excluding any values

ofx such that g(x) = u where f (u) is undefined

1 Given f : x 7! 2x + 3 and g : x 7! 1 ¡ x, find in simplest form:

In each case, find the domain of the composite function

8 a If ax + b = cx + d for all values of x, show that a = c and b = d

Hint: If it is true for allx, it is true for x = 0 and x = 1

b Given f (x) = 2x + 3 and g(x) = ax + b and that (f± g)(x) = x for allvalues of x, deduce that a = 12 and b = ¡32

c Is the result inbtrue if (g ± f)(x) = x for all x?

Sometimes we do not wish to draw a time-consuming graph of a function but wish to know

when the function is positive, negative, zero or undefined A sign diagram enables us to do

this and is relatively easy to construct

A sign diagram consists of:

² a horizontal line which is really thex-axis

² positive ( +) and negative (¡) signs indicating that the graph is above and below the

Consider the three functions given below.

Function y = (x + 2)(x ¡ 1) y = ¡2(x ¡ 1)2 y = 4

x

Graph

Notice that: ² A sign change occurs about a critical value for single factors such as

(x + 2) and (x ¡ 1), indicating cutting of the x-axis.

² No sign change occurs about the critical value for squared factors such as(x ¡ 1)2, indicating touching of thex-axis

² shows a function is undefined atx = 0

Draw a sign diagram of: a (x + 3)(x ¡ 1) b 2(2x + 5)(3 ¡ x)

a (x + 3)(x ¡ 1) has critical values b 2(2x + 5)(3 ¡ x) has critical

We try any number

> 1, e.g., x = 2

As(5)(1)

we put a+ sign here

We try any number

> 3, e.g., x = 5

As2(15)(¡2) < 0

we put a¡ sign here

As the factors are ‘single’ the signs As the factors are ‘single’ the signs

put a¡ sign here

As the factors are ‘single’

the signs alternate

We try any number> 3e.g., x = 4

As ¡4(1)2 is< 0 weput a ¡ sign here

the signs do not change

-x

+ -

-x

Draw a sign diagram for x ¡ 1

x=3

1 -\Qw_

+

-x

1 -\Qw_

GROUP INVESTIGATION SOLVING INEQUALITIES

5 are examples of inequalities.

In this section we aim to find all values of the unknown for which the inequality is true

Jon’s method of solving 3x + 2

1 ¡ x > 4 was: If

3x + 2

1 ¡ x > 4, then3x + 2 > 4(1 ¡ x)) 3x + 2 > 4 ¡ 4x) 7x > 2) x > 2

7

However, Sarah pointed out that if x = 5, 3x + 2

A graph also highlighted an error

It seems that the correct answer

is 27 < x < 1

1 At what step was Jon’s method wrong?

2 Suggest an algebraic method which does give the correct answer

-+

=1

23

From the Investigation above you should have concluded that multiplying both sides of an

inequality by an unknown can lead to incorrect results We therefore need an alternativemethod

To solve inequalities

we use these steps:

² Make the RHS zero by shifting all terms to the LHS

² Fully factorise the LHS

² Draw a sign diagram for the LHS

² Determine the values required from the sign diagram

Note: ² if a > b and c 2 R , then a + c > b + c

They concluded that there was something

wrong with the method of solution

x ¡ 4 ¡ 1 6 0)

x ¡ 1 ¡

3x

x + 1 > 0) (3x + 1)(x + 1) ¡ 3x(x ¡ 1)

(x ¡ 1)(x + 1) > 0

(x ¡ 1)(x + 1) > 0Sign diagram

of LHS is:

Thus, x 2 ] ¡ 1, ¡17[ or ]1, 1[ Example 12

f x + 22x ¡ 1 <

12

The modulus of a real numberx is its distance from 0 on the number line

So, the modulus of 7 is 7, which is written as j 7 j = 7

and the modulus of ¡5 is 5, which is written as j ¡5 j = 5:

Thus, j x j is the distance of x from 0 on the number line

THE MODULUS FUNCTION

By replacingj x j with x for x > 0 and ¡x for x < 0, write the following

functions without the modulus sign and hence graph each function:

branch y= -x2

EXERCISE 1F.1

4 Copy and complete:

What do you suspect?

5 Use the fact that j x j =px2 to prove that:

a j ab j = j a j j b j b

¯

¯ab

From the previous exercise you should have discovered these properties of modulus:

² j x j > 0 for all x ² j¡x j = j x j for all x

² j x j2= x2 for all x ² j xy j = j x j j y j for all x and y

¯ = j x jj y j for allx and y, y 6= 0 ² j a ¡ b j = j b ¡ a j for all a and b.

It is clear that j x j = 2 has two solutions, x = 2 and x = ¡2

In general, if j x j = a where a > 0, then x = §a

We use this rule to solve modulus equations

Also notice that if j x j = j b j then x = §b.

) 3x + 2 = 4 ¡ 4x) 7x = 2) x = 2

7

If 3x + 2

1 ¡ x = ¡4then 3x + 2 = ¡4(1 ¡ x)) 3x + 2 = ¡4 + 4x) 6 = x

Example 17

Notice that if j x j < 2 then x lies between ¡2 and 2i.e., if j x j < 2, then ¡2 < x < 2:

2 or x 6 ¡1

2 i.e., x 2 ] ¡ 1, ¡1

2] or [7

2,1 [Notice thataandbabove could be solved by a different method

In j 3x ¡ 7 j < 10, we see that both sides are non-negative) j 3x ¡ 7 j2 < 100 fsquaring both sidesg

) (3x ¡ 7)2¡ 102 < 0 fjaj2= a2 for allag

) (x + 1)(3x ¡ 5) > 0The critical values are x = ¡1, 53 with sign diagram:

Sometimes a graphical solution is easier.

4 Graph the functionf (x) = j x j

x ¡ 2, and hence find all values ofx for which

P, Q and R are factories which are5, 2 and 3 km away from factory O respectively

A security service wishes to know where it should locate its premises along AB so

) x < 0 or x > 2,i.e., x 2 ] ¡1, 0 [ or x 2 ] 2, 1 [ 2

=x y