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
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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  |f(x)| where f(x) is linear
19
1.7 Graphs of y  |f(x)| where f(x) is quadratic
22
Chapter 2 Quadratic Functions
28
2.1 The quadratic function
30
2.2 Completing the square
34
2.3 The quadratic formula
38
2.4 Intersection of a line and a curve
42
2.5 Quadratic inequalities
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
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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
Answers
360
Glossary
394
Index
399
Contents
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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
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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 y5 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
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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
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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
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Range
5
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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
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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
Advice and Tips
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
Advice and Tips
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
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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
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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
Advice and Tips
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
0? Give a reason for your answer.
Chapter 1: Functions
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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
Advice and Tips
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)
18 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
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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
Advice and Tips 2
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
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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
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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
2x
x +3
2f
x +3
c
2
28
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
1x
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
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