# IGCSE Additional Maths Student's Book Preview

Cambridge IGCSE®

Additional Maths STUDENT’S BOOK Also for Cambridge O Level

Su Nicholson, Peter Ransom, Carol Roberts, Trevor Senior, Brian Speed, Colin Stobartor, obart

2/12/18 5:52 PM

Contents How to use this book

vi

Chapter 1 Functions

2

1.1

4

Mappings, functions and notation

1.2 Composite functions

10

1.3 Inverse functions

11

1.4 Graphs of a function and its inverse

14

1.5 Modulus functions

17

1.6 Graphs of y ď&amp;#x20AC;˝ |f(x)| where f(x) is linear

19

1.7 Graphs of y ď&amp;#x20AC;˝ |f(x)| where f(x) is quadratic

22

28

30

2.2 Completing the square

34

38

2.4 Intersection of a line and a curve

42

45

Chapter 3 Equations, Inequalities and Graphs

50

3.1 Solving absolute-value linear equations

52

3.2 Solving absolute-value linear inequalities

57

3.3 Solving cubic inequalities graphically

64

3.4 Graphs of cubic polynomials and their moduli

68

3.5 Solving quadratic equations by substitution

72

Chapter 4 Indices and Surds

78

4.1 Simplifying expressions with negative, zero and fractional indices

80

4.2 Solving equations with indices

82

4.3 Surds

84

4.4 Rationalising the denominators of surds

86

4.5 Solving equations with surds

88

Chapter 5 Factors and Polynomials

94

5.1 The factor theorem

96

5.2 The remainder theorem

102

Contents

iii

2/15/18 10:11 AM

Chapter 6 Simultaneous Equations

106

6.1 Simultaneous equations

108

6.2 Interpreting and solving simultaneous equations graphically

111

Chapter 7 Logarithmic and Exponential Functions

118

7.1

Properties of exponential functions and their graphs

120

7.2 Properties of logarithmic functions and their graphs

123

7.3 Laws of logarithms

128

7.4 Changing the base of a logarithm

131

7.5 Equations of the form a  b

132

x

Chapter 8 Straight-Line Graphs 8.1 Interpreting equations of the form y  mx

136 c

138

8.2 Transforming relationships of the form y  ax to linear form

143

8.3 Transforming relationships of the form y  Ab to linear form

147

8.4 Working with the mid-point and length of a straight line

151

8.5 Working with parallel and perpendicular lines

154

n

x

Chapter 9 Circular Measure 9.1 Radians

166

9.2 Arc length

167

9.3 Sector area

169

9.4 Problems involving arcs and sector area

171

Chapter 10 Trigonometry

180

10.1 Trigonometrical values for angles of any magnitude

182

10.2 Further trigonometrical functions

192

10.3 Other trigonometrical functions

194

10.4 Graphs of trigonometric functions

196

10.5 Trigonometric identities

207

10.6 Solving trigonometric equations

209

Chapter 11 Permutations and Combinations

216

11.1 Permutations

218

11.2 Combined permutations

220

11.3 Combinations

224

11.4 Problems with permutations and combinations

226

Chapter 12 Series

iv

164

234

12.1 Arithmetic progressions

236

12.2 Geometric progressions

239

Contents

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12.3 Inﬁnite sequences

242

12.4 Binomial expansions

244

Chapter 13 Vectors

252

13.1 Position and unit vectors

254

13.2 Vectors in geometry

258

13.3 Compose and resolve velocities

262

Chapter 14 Differentiation

272

14.1 The idea of a derived function

274

14.2 Differentiating polynomials

277

14.3 Differentiating trigonometric functions

281

14.4 Differentiating exponential and logarithmic functions

285

14.5 Differentiating products of functions

290

14.6 Differentiating quotients of functions

292

Chapter 15 Applications of Differentiation

298

15.1 Calculating gradients, tangents and normals

300

15.2 Stationary points

302

15.3 Small increments and approximations

305

15.4 Practical applications

307

Chapter 16 Integration

314

16.1 Integration as anti-differentiation

316

16.2 Integrating polynomials

321

16.3 Integrating trigonometric functions

324

16.4 Integrating exponential and other functions

326

16.5 Evaluating deﬁnite integrals

330

16.6 Integrating to evaluate plane areas

334

Chapter 17 Applications of Kinematics

342

17.1 Applying differentiation to kinematics

344

17.2 Applying integration to kinematics

347

17.3 Using x–t and v–t graphs

350

360

Glossary

394

Index

399

Contents

v

2/20/18 7:40 PM

1 Functions

C H A P T ER

Topics

Key worrds

1..1 Ma Mapp pings ingss, fu in unc nc tion tion ti ns and d no nota tatiio on n

m pp ma ppin in ng diiag gra ram, m, one ne-o -one -o ne,, many-on ne ne, one-many, man anyy-ma yman ma ny, fu ny unc ncti c tiion, on, do on doma main ma a and ran an nge

1..2 Co omp positte fu uncc tion on s on

comp co mpos mp osit os itte fu func nc ti nct tion

1..3 Inve 1 vve ersse fu fun ncc tiion ns

inve in nve vers r e ffu unc ti tio on, se elf l -i -inv nver nv erse fu un nct ction

1..4 G 1 Grrap phs hs of a fu func tion tion and ti d its ts in nvve errse r e 1.5 1. 5 Mo od du ullu us fu uncc ti tion on o ns

mo odu d lus, lus, abs lu bsol ollut ol utte fu u func n ti nc t on n

1.6 G Graphs hss of y  |f ( x)| )| whe ere e f ( x) iiss linea in nea earr 1 7 Grra 1. ap phs phs of y  |f ( x))|| whe h rre e f ( x) iss qua uadr drat dr a icc at

ro oot ots, s,, turrni ning ng p o oiin ntt, st stat attio ona n ryy poi ointt

In this chapter you will learn how to • Ap A pl plyy an a d un unde nde ders ers rsta t nd th he e terrmss: fu uncttio tion, on, do doma maiin, rang rang ge (im (imag mage set), one e -one ne fun u cttion,, in nve v rrsse fu func ncc tion tion o , comp cco o positte fu fun ncc tiion • Co Corr r ectl rr tly tl y us use e tth he notati tatiion ta on f (xx)  2 x 2

3, f : x

7x

2, f 1( x) an nd f 2( x) x  fff ( x)

• Re R co cogn cogn g ise an and un nde ders ers rssttta an nd d th he e relat elatio ons nship p betw betw twee e n y  f ( x) an ee and y  |ff (x) x)|,, wh her ere e f ( x) ma ay be e linear or qu uad a rati tiic • Fiind d out u ho ow w to o fo form form r and d sol olve compl olv ple ex com ex omp om posite e fun nct c tio ons • Diisco scoverr why a giv iven ven n fu un nct c tio tio ion n iss a fun unct unct c tio ion io on or why it does do oes not hav ave an an inv nver nver erse se • Fi Find out utt how w to ﬁnd d the e inv n errse s of a on onee-on eo e func on funcc ti fu tion on n • Us Use e sk ketch h gra r ph hs tto o show how th he re ela l ti t on nsh hip p be ettwe een a fun unct cttion io a io an nd it i s in i ve vers r e. rs

From your Cambridge IG GCSE Mathematics course you should be ablee to o •

Re eco cogn gnis gn i e th is the he no n ta ati t on n that is is usse ed to ed o desscrib be a fu uncti ncc ti tion on n

Find Fi nd the he e in nv ver verrsse es of o simp imple e fu unct c tion ons on ns

Form Fo m com mpo osi s te e fun unct c io ct ions ns ns

Skettch Sk h and rec e og ogni nis ni ise the he gra aph phs of a linea phs ar a an nd qu ua ad drra ati tic e eq qua uati tiion ons ns

Sk kettch c and d rec eccog ecog gnise the gra aph p s fo for si s n x an and co and co oss x

2/3/18 1:50 PM

Starting point Thiiss top Th This opiic exten nds what you have learned about functtiio ons n in Ca Camb m ri r dg ge IG GCSE CSSE Ma ath t em mat atics. s. Be efo f re r you u movve on to look in more detail at functionss, re evi view w wha hat yo you al alre ead adyy kn know ow o w abo out u evval alua uatting ua g funct ctio ct onss and ﬁnding an inverse function. 2 f ( x)  , x ≠ 0 g(x) x  3 2x h(x) x  x2 1 x 1 Fi F nd n fg g⎛ 1 ⎞ 3 Find g 1( x) ⎝ 2⎠ 4 Solve g(xx)  3x 2 Fi F nd the value e of x when h(x) x  35

W hy this cha apteer matters Funcc ti Fu tio ons al ons on allo llow lo ow yo ou to to describe many of yyo our ur eve very r da day ac acti tion ti ion ons al alg ge g ebr b ai aica c llly. y Whe heth her e you o are e calc ca lcul lc ulat ul attin ing g th the costt of post pos ing a parcel or wo po ond derin errin ng ho how ow lo long ong n it w wiill l tak ak a ke to t dow wnl n oa oad ad a ﬁ ﬁle e accro r ss the h in ntter ernet,, fun ern u cttions are likelyy tto o be in invo vo olv l ed e . All Al th Al thes ese exxam ese es ampl p es es inv nvo nv ollve ve an in nde dep depe pe end nden entt orr inp n ut val alue u (a pa ue p rccel e ’ss mass, the ﬁle e si size z ) an nd a de epe p nden nd den nt or outtpu ut va valu lu ue (c ( os ostt of o possting tiing ng , time ti me e tak aken en). ).. In ea each c casse,, these ch se e value ue es a arre li link nk ked e by a fu uncc ttiion n. A imp An m or orta tant pro rop ope pert per r t y of rty of a fun unct ctio io on iss tha h t ea e cch h inp np u utt val a ue e pro odu duce ces ce es on only y one e ansswe wer. r Thi hs is sen is nsi s ble. Iff a fu unc n ti t on to oo ok an inputt val alue lue and d pro rod duce duce ed a va vari riet et y of of ans n wer wers, rs, o orr a dif iffe f fe fere rent re nt nt a sw an wer each ti t me the e cal a cu c lation o was a carriied e out, ho how us usef eful ull wou ould d it be be? 1 1  2 wou ould ld d be tr true e only on nly y som ome me o off the he e time, e, nott alway ayys! Lear Le arrni ning ing g how to o crea e te and use ea se funct ctio tio ions ns is al also so an im so impo port po rtan rt antt st an step eppi ping ing sto tone e on yo your urr mat athe hema mati tica call jo our u ne n y,, sin ince ce the here he e are e som me th things gs tha hat ca ha can on only ly y be done done e witth fu unctions, s for o exa ample mp ple e, diiff f er eren e ti t attio on and an d in nte egrat grattion whic ich yo ic you will mee et in n Cha apt pter er 14 of o this bo book ok. ok Leon Le on nha hard rd Eul u er (1707 0 –8 83) was the ﬁrst st pe erson erso n to o use e f ( x) to rep epre resent a func fu ncc tion on n of xx. He e was a Swiiss ss mat a he hema ma ati t ciian who o obta btaine n d a mast ste er’s deg egre re ee in n mathe athe ema mattics at th he ag age of o 16. 6 Alttho ough h Eu Eule le er be eca ame me bli l nd at ag age e 59 59,, it i did d nott st no stop op him m fro om in nvest stigat st atin at i g and d wrriittin ng on a wid ide e ra r ng ge off sci c en enti t ﬁc an a d m th ma hemat ematiccal a top opicss – a tota op al off 886 6 books and d pamph phle le ets ts..

Exploring the topics A s you As ou sta tart rtt to ex e pllor o e tth hiss top picc, di d scus usss po p ssible answers to the folllo ow win ng q qu ue essti tion ns in pairs. 1 How migh g t a pa gh parc r ell del eliv iv ver erin in ing ng bu ussiine nesss use e fun nctio onss to ca c lccu ullat ate te the th he co ost of delive veriin ve ng ga parcel when n va v riiable lle es su succh h as we weig weig igh htt an nd d dis ista stanc an nccce e ar are e in nvo v lv ved ed? 2 Using a grap phi h ca c l ca alccu ullator atorr or g at grrap aphing hing hi g app pp, pllot ot th he grra he aph p so off the hese ese e paiirs rs of e eq q qua uation uati ua tio ti on ns in in the same e gra ra aph h spa pace. ce e. x2 x −3 a y  2xx 3 and nd y  b y5 and d y  15 f r 0 x 5 in bo otth h casses es)) 15 − 3 x (fo 3 2 In each pa aiir, in pa partts a and d b, wh whe e iiss the wher he lin ine of of sym ymme metr tr y b be etw t ee een en th he tw two o gr grap aphs ap phs hs? s? 1 c Drraw the gra aph h of y  . Wh Wher ere d er do o you ou thi hink k the he re eﬂ ﬂectio c tion ct n off this gr grap ap ph wi willl ll be? e? x

Chapter 1: Functions

3

2/7/18 11:23 AM

Chapter 1 . Topic 1

1.1 Mappings, functions and notation Mappings Two data sets may be connected, or related, by a rule. For example, in a physics experiment the extensions of a spring (cm) when various masses (kg) were added were recorded. The results are shown in this diagram. It was noted that the connection between these two sets of data 4 mass added. is: extension  3 The rule connecting these data sets ‘Extension’ and ‘Mass’ is: 4 ‘multiply the mass by ‘. 3 This type of diagram is called a mapping diagram. The starting data set is the domain and the resulting data set is the range.

Weight (kg)

Extension (cm)

0.75 1.5 3.75 7.5 15

1 2 5 10 20

Domain

Range

Each mass value produces only one extension value. If ‘Mass’ is the input value (x) and ‘Extension’ the output value, you can write the rule algebraically as: 4 x x 3 4 Read this as ‘x is mapped to x’. 3 You can plot mappings, one against the other, to produce the graph, in this case it is a linear graph. 10

Extension (cm)

8 6 4 2 0

0

1

2

3

4 Mass (kg)

5

6

7

8

This example, where for each input value there is only one output value, is a one-one mapping. Mappings may also be many-one, one-many and many-many. A many-one mapping will have more than one input value for each output value.

4

1.1 Mappings, functions and notation

2/3/18 1:51 PM

1.1 An example is x

x2 y

x

x2

y ⫽ x2

4

y

–2 –1.5 –1 0 1 1.5 2

4 2.25 1 0 1 2.25 4

Domain

Range

3 2 1

–3

–2

–1

0

0

1

2

3

x y

When graphed, this produces the familiar parabola of a quadratic graph.

3

A one-many mapping will have one input value producing two output values.

2

x

An example is x

1

In many-many mappings each input value has more than output value, but each output value also has more than one input value. r −x 2

An example is a circle: x

y

2

0

0

1

2

x

3

–1

y ⫽ 冪莥 4⫺x –2

1

–1

0

2

2

–2

y ⫽ ⫾冪莥x

–3 0

1

2

x

–1 –2

Functions A function is a rule, or relation, in which each input value (x) produces, or maps, onto a single output value (y). If there is more than one value of y for the same value of x, the rule is not a function. This means that only one-one and many-one mappings are functions, because in one-many mappings one input value can result in many output values, and in many-many mappings there are multiple inputs and outputs.

Advice and Tips Geometric test for functions: on the graph, draw vertical lines parallel to the y-axis. If any of the lines meet the curve at more than one point, the graph is not a x. function, as in y  –1

0

1

2

3

4

–1

0

1

2

3

4

Domain

The domain of a mapping is the set of the input values, also called the independent variables. The range of a mapping is the set of values, also called the dependent variables, which are the result of applying the rule.

5

6

Chapter 1: Functions

Range

5

2/3/18 1:51 PM

If the solution to a problem involves more than one function you would normally use f(x) to represent the ﬁrst function, then a different identiﬁer for the second function, such as g(x). Generally, the letters f, g and h are used for the purposes of multiple functions but in fact you can use any letter.

Trigonometric functions You are familiar with the graphical representations of the three trigonometrical ratios of sine (sin), cosine (cos) and tangent (tan). y 1 y = Sine θ

0.7071

–90°

0

0

45°

90°

135°

180°

270°

360°

θ

–1 y 1 y = Cosine θ

–90°

0

135° 0

90°

180°

270°

360°

θ

–0.7071 –1 y 2

y = tangent θ 1

–90°

0

0

90°

135°

180°

270°

360°

θ

–1

–2

6

1.1 Mappings, functions and notation

2/3/18 1:51 PM

1.1 These are also functions. Each angle has its own speciﬁc cos, sin and tan values. You will consider these functions in detail in Chapter 10.

Notation From the above examples: 4 x x where x 3 x x2 where x

ℝ, is a one-one mapping

and they can be written as: 4 f: x x x ℝ 3 g: x x2 x ℝ

or, alternatively as: 4 f(x)  x x ℝ 3 g(x)  x 2 x ℝ

If f(x)  2x

f(5)  2(5)

3, then

ℝ is ‘the set of real numbers’, means ‘is a member of’.

ℝ, is a many-one mapping

f( 3)  2( 3)

3  13, and

3 3

Read f(x) as ‘f of x’. It is sometimes called the image of x.

So since f(5)  13 and f( 3)  3, the points (5, 13) and ( 3, 3) lie on the graph. y

Example 1

8

This is the graph of f(x). State its domain and range.

(3, 6)

6

Solution

4

The minimum and maximum values of x are 2 and 7 so the domain is 2 x 7

2

The minimum and maximum values of y are 2 and 6 so the range is 2 y 6

0

–4 –2 (–2, –2)

(7, 0) 0

2

4

6

8

x

–2 –4

Example 2 x

3

2x

x

ℝ, 4

x

3

a State what kind of mapping this is and whether it can be deﬁned as a function. b If the mapping is a function: i

state the domain of the function

ii write it as a function, using the correct notation iii sketch a graph of the function iv determine the range of the function.

Chapter 1: Functions

7

2/3/18 1:51 PM

Solution a This is a one-one mapping and is a function. b i

The domain is: 4

ii f(x)  3

x

2x, or f: x

iii When x  4, y  3 y  3 2(3)  3

3 3

2x

2( 4)  11 and when x  3,

The line has a y-intercept of 3 y 12 (–4, 11)

10 8 6 4

(0, 3)

2

–6

–4

–2

0

0

2

4

x

–2 –4

(3, –3)

iv The minimum and maximum values of y are 3 and 11 so the range is 3

y

11

Exercise 1.1 1

Identify the type of mapping for each function. a f: x

2x

d f(x)  5 2

1 0.5x

b f(x)  x 2

c f(x)  3x

e f: x

f

sin(x)

f: x

x3

2x 2

1

For each graph, work out the domain and range and decide whether it is the graph of a function. y

a

y

b

(–3, 8)

(–2, 4)

(3, 4)

(2, 4) 2 0

8

(1, 2) 0

0 x

0

x

1.1 Mappings, functions and notation

2/3/18 1:51 PM

1.1 y

c

y

d

4

0

0

x

2

–4

0

0

4

x

–4

y

e

0

x

0

(–2, –1)

3

a f(x)  5x

3, 4

c f: x

2)2

(x

x

2

b g(x)  x 2

3, x

0

d g: x

e f(x)  2 + 4 + x , 4 2

PS

4

Find the range for each function.

x

x

2, 2 x 4 1 , 3 x 3, x ≠ 0 x

4

Draw a sketch of each function ﬁrst to help visualise the graph.

The population (P) of a species of bird in a region is declining rapidly. The calculation P(t)  2400

180t gives the population in t years from now.

a What is the population of birds now? b Find P(5) and explain what this means. c Find t when P(t)  960 and explain what this represents. d Is the function valid for all t

Chapter 1: Functions

9

2/3/18 1:51 PM

Chapter 1 . Topic 2

1.2 Composite functions In this section, you will use the notation f(x) to deﬁne a function. Consider the two functions: f(x)  (x

2)2 and g(x)  3x

Remember that f 2  f so that f 2(x)  ff(x) 101010101010

If you evaluate f(4) as 36 and then used this as the input to g(x), g(36), the result is 108.

f,

This combination of f(x) and g(x) is written as gf(x). In cases like this, where the output from one function is used as input to another, the result is a composite function. A

B

C

1 2 3 4

9 16 25 36

27 48 75 108

f(x) ⫽ (x ⫹ 2)2

g(x) ⫽ 3x

gf(4)  108 but note that fg(4)  196

A

C

1 2 3 4

27 48 75 108

From these two examples, you should see that you must carry out the right-most function followed by the next function to the left.

gf(x)

Example 3 f(x)  2x 2

4 and g(x)  3x

1

Find: a fg(4)

b gf(5)

c

g2(3)

Solution a g(4)  3(4)

1  11

f(11)  2(11)2

b f(5)  2(5)2 g(46)  3(46)

4  238

4  46

c g(3)  3(3)

1  137

g(8)  3(8)

18 1  23

You can create a single expression for a composite function.

Example 4 f(x)  2x 2

4 and g(x)  3x

1

Find fg(x) and gf(x)

Solution fg(x)  f(3x  2(3x

10

1) 1)2

4

gf(x)  g(2x 2

4)

 3(2x 2

4)

 2(9x 2

6x

1)

4

 6x 2

12

 18x 2

12x

2

4

 6x 2

13

 18x 2

12x

2

1 1

1.2 Composite functions

2/3/18 1:51 PM

1.3 Check from the example above: fg(4)  18(4)2

12(4)

gf(5)  6(5)2

13  137

2  238

Exercise 1.2 1

f(x)  2x 2

2 and g(x)  5

a Find:

i

3x.

fg(3)

ii gf(4)

iii

fg(x)

iv

gf(x)

b Use your answers to a i and ii to check a iii and iv. 2

3 and g(x)  3x

f(x)  x 2 a f(5) e

3

f 2(2)

f(x)  x 2

4x

1. Find:

b g(6)

c

fg(4)

d

gf(3)

f

g gf(x)

h

f 2(x)

fg(x)

3 and g(x)  2x

a fg(x) in its simplest form 4

f(x)  (x

5

h(x) 

2)2

a gh(x) 6

f(x)  x 2

E

7

f(x) 

b

fg(2)

c

fg(0)

d fg( 2)

2x − 2 . Find fg(6). x−4 4. Find:

3 and g(x) 

x , and g(x)  2x

E

1. Find:

b

hg(x)

2x and g(x)  x

3. Solve the equation fg(x)  0.

2x − 2 and g(x)  2 x − 1. Solve the equation fg(x)  4. x−4 3

1.3 Inverse functions In this section, the notation f(x) is used to deﬁne a function. The inverse of a function f(x) is the function that has the opposite, or reverse, effect of the original and is written as the inverse function f 1(x). The function f(x)  4x

2 ‘multiplies x by 4 and adds 2’.

The reverse function will ‘subtract 2 and then divide by 4’: x − 2 , so f 1(10)  2 and f 1(4)  0.5 f 1(x)  4 To ﬁnd the inverse of a function: Write y  f(x) y  4x

then rearrange to make x the subject: y − 2 y 2  4x x 4

Given f(n), then f 1f(n)  n, which allows you to check that you have created the inverse correctly.

Chapter 1: Functions

11

2/3/18 1:51 PM

Chapter 1 . Topic 3 Finally, replace y in the answer with x: f 1(x)  Consider the graph of the function f(x)  x 3.

x − 2 4

Given a value in the range, such as y  3, it is possible to ﬁnd the corresponding member of the domain. To do this, you draw a horizontal line on the graph at y  3 and then, where the line meets the curve, draw a line down to the x-axis. In this case x ≈ 1.4, a member of the domain. This is the opposite, or inverse, of the suggestion for a geometric test above. For each value in the range of f(x)  x3 there will be only one corresponding value in the domain.

y y⫽3

3 2 1 0

–1

y

Not every function has an inverse.

x

y ⫽ x2

5

Consider f(x)  x 2. It has a many-one mapping and so is a function. Given the value y  4, and drawing a horizontal line (as above for f(x)  x 3), you can see that there will be two domain values, 2 and 2.

4 3

Attempting to ﬁnd f 1(4) gives 2. This conﬂicts with the idea that there is only one domain value, and tells you that f(x)  x 2 does not have an inverse.

2

Only those functions with a one-one mapping have an inverse. When you draw the graphs of f(x) and f 1(x) you will see that they are reﬂections of each other in the line y  x.

2

–1

This indicates that the function f(x)  x 3 has an inverse.

1

–3

In general, if f(x)  y, then f 1(y)  x and vice versa, if f 1(y)  x then f(x)  y. Some functions are the same as their inverse functions. For example, if f(x) 

1 x ⫽ 1.4

0

1 1 for x ≠ 0, its inverse is f 1(x)  . These x x

special cases are self-inverse functions.

–2

–1

0

0

1

2

3

x

Advice and Tips The domain of f 1(x) is the range of f(x). The range of f 1(x) is the domain of f(x).

If you draw the graph of a self-inverse function you will ﬁnd that it is symmetrical about the line y  x. Always take care when ﬁnding the inverse of a composite function. If fg(x) then its inverse is g 1f 1(x). Note that g and f are reversed.

Example 5 f(x)  (x

2)2

3 for x

1

a Find an expression for f 1(x)

12

b

Find f 1(13)

c Solve the equation f 1(x)  8

1.3 Inverse functions

2/3/18 1:51 PM

1.3 Solution a y  (x

2)2

3  (x

y

b f 1(x) 

3

x +3

2)2

f 1(13)  13 + 3

2

f 1(13)  16

y +3  x y +3

2x

x +3

2f

x +3

c

2

28

x + 3  10

2

( x + 3)2  100

2

f 1(13)  2

x

1(x)

3  100

x  97

Example 6 f(x)  (x

1)2

3 for x

1 b Find the value of x when f 1(x)  f( 1)

a Find an expression for f 1(x)

Solution a y  (x

1)2

b f( 1)  ( 1

3

3  (x

1)2

y −3  x

1

3

x −3

y −3

1x

4

x −3

f 1(x) 

x −3

y

1)2

3

f( 1)  3

16  x

1

1

3

19  x f 1(x)  f( 1) when x  19

Exercise 1.3 For each of the functions in questions 1–5, ﬁnd an expression for f 1(x). 1

f(x)  (x

2

f(x) 

3 4 5 6

2)2

4

3x + 1 4 8 − 3x f(x)  5x 5 f(x)  ( x − 32) 9 f(x)  ( x − 2)3

E

3

2x x2 − 2 d 3

x Show that f(x)  is a self-inverse x −1 function.

8

f(x)  x 2 If x

E

9

3, and g(x)  5x

1

0 solve fg(x)  g 1(34)

f(x)  2x

3 and g(x)  (x

2)2

Find fg(x) and its inverse.

Which of these functions have no inverse? 3 2 x 5 a f(x)  5 − x 2 b f(x)  3x x 3 2 x 2 c

7

x

0

4

x

E

10

f(x)  ax

b (a

0) and f 2(x)  16x

Find: a the values of a and b b f 3(x)

4

Chapter 1: Functions