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lIGCSEAlPast Year
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Unit 02
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Functions
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Unit 02 Functions
0606 Additional Mathematics
2
Mathematical Formulae
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1. ALGEBRA
Quadratic Equation
For the equation ax2 + bx + c = 0,
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x=
l
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A
2
b − 4 ac
2a
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Binomial Theorem
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n
n
n
(a + b)n = an + 1 an–1 b + 2 an–2 b2 + … + r an–r br + … + bn,
n
n!
where n is a positive integer and r =
(n – r)!r!
2. TRIGONOMETRY
Identities
sin2 A + cos2 A = 1
sec2 A = 1 + tan2 A
cosec2 A = 1 + cot2 A
Formulae for ∆ABC
a
b
c
sin A = sin B = sin C
a2 = b2 + c2 – 2bc cos A
∆=
1
bc sin A
2
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2! of !21
0606/12/F/M/15
Unit 02 Functions
0606 Additional Mathematics
14
0606/21/M/J/14
12 The functions f and g are defined by
f ^xh =
g ^xh =
2x
for x 2 0 ,
x+1
(i) Find fg ^8h.
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x + 1 for x 2-1.
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[2]
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ax
, where a, b and c are integers
(ii) Find an expression for f 2 ^xh, giving your answer in the form
bx + c
to be found.
[3]
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(iii) Find an expression for g ^xh, stating its domain and range.
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[4]
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3! of !21
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Unit 02 Functions
0606 Additional Mathematics
15
0606/21/M/J/14
-1
12 (iv) On the same axes, sketch the graphs of y = g ^xh and y = g ^xh, indicating the geometrical
relationship between the graphs.
[3]
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4! of !21
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Unit 02 Functions
0606 Additional Mathematics
12
0606/22/M/J/14
11
The functions f and g are defined, for real values of x greater than 2, by
f (x) = 2 x - 1,
y
g (x) = x ^x + 1h.
(i) State the range of f.
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[1]
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(ii) Find an expression for f -1 (x) , stating its domain and range.
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[4]
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5! of !21
0606/22/M/J/14
Unit 02 Functions
0606 Additional Mathematics
13
0606/22/M/J/14
11 (iii) Find an expression for gf (x) and explain why the equation gf (x) = 0 has no solutions.
[4]
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6! of !21
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[Turn over
Unit 02 Functions
0606 Additional Mathematics
16
0606/23/M/J/14
12 The function f is such that f (x) = 2 + x - 3 for 4 G x G 28 .
(i) Find the range of f.
y
(ii) Find f 2 (12) .
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(iii) Find an expression for f (x) .
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The function g is defined by g (x) =
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120
for x H 0 .
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(iv) Find the value of x for which gf (x) = 20 .
A
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Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every
reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the
publisher will be pleased to make amends at the earliest possible opportunity.
University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of
Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
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© UCLES 2014
7! of !21
0606/23/M/J/14
Unit 02 Functions
0606 Additional Mathematics
5
0606/21/O/N/14
4
The functions f and g are defined for real values of x by
f (x) =
g (x) =
for x 2 1,
x-1-3
x-2
2x - 3
(i) Find gf(37).
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(ii) Find an expression for f - 1 (x) .
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(iii) Find an expression for g -1 (x) .
[2]
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© UCLES 2014
y
for x 2 2.
8! of !21
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[Turn over
Unit 02 Functions
0606 Additional Mathematics
10
0606/23/O/N/14
7
The functions f and g are defined for real values of x by
f ^xh =
2
+ 1 for x 2 1,
x
g ^xh = x 2 + 2 .
Find an expression for
f -1 ^xh,
(i)
(ii)
gf ^xh,
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fg ^xh.
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(iii)
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9! of !21
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Unit 02 Functions
0606 Additional Mathematics
11
0606/23/O/N/14
7
(iv) Show that ff ^xh =
3x + 2
and solve ff ^xh = x .
x+2
[4]
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! of !21
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[Turn over
Unit 02 Functions
0606 Additional Mathematics
5
0606/12/F/M/15
3
(i) On the axes below sketch the graph of y = 4 - 5x ,
where the graph meets the coordinate axes.
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(ii) Solve
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4 - 5x = 9 .
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[3]
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stating the coordinates of the points
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[3]
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! of !21
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[Turn over
Unit 02 Functions
0606 Additional Mathematics
10
0606/12/F/M/15
8
f (i) = sin 2i for 0 G i G
(a) A function f is such that
r
2
.
(i) Write down the range of f.
[1]
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(ii) Write down a suitable restricted domain for f such that f -1 exists.
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(b) Functions g and h are such that
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g (x) = 2 + 4 ln x for x 2 0 ,
h (x) = x 2 + 4 for x 2 0 .
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(i) Find g -1 , stating its domain and its range.
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(ii) Solve gh (x) = 10 .
[3]
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! of !21
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0606/12/F/M/15
Unit 02 Functions
0606 Additional Mathematics
12
0606/22/M/J/15
f: x 7 sin x
10 (a) The function f is defined by
graph of y = f (x) .
for 0° G x G 360° . On the axes below, sketch the
[2]
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90°
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180°
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(b) The functions g and hg are defined, for x H 1, by
ca
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(i) Show that h ^xh =
A
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y
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360°
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g ^xh = ln ^4x - 3h,
hg ^xh = x.
ex + 3
.
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Unit 02 Functions
0606 Additional Mathematics
13
0606/22/M/J/15
(ii)
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y = g(x)
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x
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The diagram shows the graph of y = g ^xh. Given that g and h are inverse functions, sketch, on the
same diagram, the graph of y = h ^xh. Give the coordinates of any point where your graph meets
the coordinate axes.
[2]
ca
(iii) State the domain of h.
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(iv) State the range of h.
[1]
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[1]
0606/22/M/J/15
! of !21
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[Turn over
Unit 02 Functions
0606 Additional Mathematics
10
0606/13/M/J/15
8
f ^xh = 3e 2x for x H 0 ,
g ^xh = ^x + 2h2 + 5 for x H 0 .
It is given that
(i) Write down the range of f and of g.
(ii) Find g -1 , stating its domain.
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(iii) Find the exact solution of gf ^xh = 41.
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! of !21
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0606/13/M/J/15
Unit 02 Functions
0606 Additional Mathematics
11
0606/13/M/J/15
8
(iv) Evaluate f l^ln 4h.
[2]
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! of !21
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[Turn over
Unit 02 Functions
0606 Additional Mathematics
4
0606/23/M/J/15
2
(a)
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The diagram shows the graph of y = f (x) passing through ^0, 4h and touching the x-axis at ^2, 0h.
Given that the graph of y = f (x) is a straight line, write down the two possible expressions for f (x) .
[2]
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(b) On the axes below, sketch the graph of y = e -x + 3, stating the coordinates of any point of
intersection with the coordinate axes.
[3]
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x
! of !21
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Unit 02 Functions
0606 Additional Mathematics
14
0606/12/O/N/15
11
(a) A function f is such that f ^xh = x 2 + 6x + 4 for x H 0 .
(i)
(ii)
Show that x 2 + 6x + 4 can be written in the form ^x + ah2 + b , where a and b are integers.
[2]
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Write down the range of f.
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(iii)
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em
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Find f - 1 and state its domain.
in
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Unit 02 Functions
0606 Additional Mathematics
15
0606/12/O/N/15
11
(b) Functions g and h are such that, for x d R ,
g ^xh = e x
h ^xh = 5x + 2 .
and
y
Solve h 2 g ^xh = 37 .
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Question 12 is printed on the next page.
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© UCLES 2015
0606/12/O/N/15
! of !21
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[Turn over
Unit 02 Functions
0606 Additional Mathematics
8
0606/13/O/N/15
6
y = x 2 - 4x - 12
(i) On the axes below, sketch the graph of
pointswherethegraphmeetstheaxes.
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y = x 2 - 4x - 12 .
[2]
A
(iii) Findthevaluesofksuchthattheequation
x 2 - 4x - 12 = k
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©UCLES2015
y
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a
c
(ii) Findthecoordinatesofthestationarypointonthecurve
y
showing the coordinates of the
[3]
hasonly2solutions.
[2]
! of !21
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Unit 02 Functions
0606 Additional Mathematics
11
0606/23/O/N/15
9
Given that f (x) = 3x 2 + 12x + 2 ,
(i) find values of a, b and c such that f (x) = a (x + b) 2 + c ,
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(ii) state the minimum value of f(x) and the value of x at which it occurs,
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(iii) solve f c m = 0 , giving each answer for y correct to 2 decimal places.
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[Turn over