MINDBRIDGE
CA IE ADDITIONAL MATHEMATICS
TOPICAL PRACTICE QUESTIONS
TOPIC 1:
FUNCTIONS
VOLUME 1:
Paper 1
Variants 1, 2 and 3
2016- 2020
"Allow yourself to be a
beginner, no one starts
off being excellent."
MINDBRIDGE
1
The function f is defined by
f (x) = 2 - x + 5
for - 5 G x 1 0 .
(i) Write down the range of f.
[2]
(ii) Find f -1 (x) and state its domain and range.
[4]
The function g is defined by
(iii) Solve fg (x) = 0 .
g (x) =
4
x
for - 5 G x 1 - 1 .
[3]
MINDBRIDGE
2
(a) It is given that f ^xh = 3e - 4x + 5 for x ! R .
(i) State the range of f.
[1]
(ii) Find f -1 and state its domain.
[4]
(b) It is given that g ^xh = x 2 + 5 and h ^xh = ln x for x 2 0 . Solve hg ^xh = 2 .
[3]
MINDBRIDGE
3
Diagrams A to D show four different graphs. In each case the whole graph is shown and the scales on
the two axes are the same.
A
B
y
y
x
O
x
O
C
D
y
y
x
O
x
O
Place ticks in the boxes in the table to indicate which descriptions, if any, apply to each graph. There
may be more than one tick in any row or column of the table.
[4]
A
Not a function
One-one
function
A function
that is its own
inverse
A function
with no inverse
B
C
D
MINDBRIDGE
4
The diagram shows the graph of a cubic curve y = f (x) .
y
y = f (x)
5
–2
–1
0
5
x
(a) Find an expression for f (x) .
[2]
(b) Solve f (x) G 0 .
[2]
MINDBRIDGE
5
(a) Functions f and g are such that, for xdR ,
f ^xh = x 2 + 3,
g ^xh = 4x - 1.
(i) State the range of f.
[1]
(ii) Solve fg (x) = 4 .
[3]
MINDBRIDGE
(b) A function h is such that h ^xh =
2x + 1
for xdR , x ! 4 .
x-4
(i) Find h -1 ^xh and state its range.
[4]
(ii) Find h 2 ^xh, giving your answer in its simplest form.
[3]
MINDBRIDGE
6
(a) f (x) = 3 - cos 2x for 0 G x G
r
2
.
(i)
Write down the range of f.
[2]
(ii)
Find the exact value of f -1 (2.5) .
[3]
MINDBRIDGE
(b) g (x) = 3 - x 2 for x ! R .
Find the exact solutions of g 2 (x) =- 6 .
[4]
MINDBRIDGE
7
f (x) = 3e 2x + 1
for x d R
g (x) = x + 1
for x d R
(i)
Write down the range of f and of g.
[2]
(ii)
Evaluate fg 2 (0) .
[2]
(iii)
On the axes below, sketch the graphs of y = f (x) and y = f - 1 (x) , stating the coordinates of the points
where the graphs meet the coordinate axes.
[3]
y
O
x
MINDBRIDGE
8
3
It is given that f (x) = 5 ln (2x + 3) for x 2- .
2
(a) Write down the range of f.
[1]
(b) Find f -1 and state its domain.
[3]
(c) On the axes below, sketch the graph of y = f (x) and the graph of y = f -1 (x) . Label each curve
and state the intercepts on the coordinate axes.
y
O
x
[5]
MINDBRIDGE
f : x 7 (2x + 3) 2 for x 2 0
9
(a) Find the range of f.
[1]
(b) Explain why f has an inverse.
[1]
(c) Find f -1.
[3]
(d) State the domain of f -1.
[1]
(e) Given that g : x 7 ln (x + 4) for x 2 0, find the exact solution of
fg (x) = 49 .
[3]
MINDBRIDGE
10 Functions f and g are defined, for x 2 0 , by
f (x) = ln x ,
g (x) = 2x 2 + 3.
(i) Write down the range of f.
[1]
(ii) Write down the range of g.
[1]
(iii) Find the exact value of f -1 g (4) .
[2]
(iv) Find g -1 ^xh and state its domain.
[3]
MINDBRIDGE
11 (a) It is given that
f :x7 x
g : x 7 x+5
for x H 0 ,
for x H 0 .
Identify each of the following functions with one of
f -1, g-1, fg, gf, f 2, g2.
(i)
x+5
[1]
(ii)
x-5
[1]
(iii) x2
[1]
(iv)
[1]
x + 10
(b) It is given that
h (x) = a +
b
where a and b are constants.
x2
(i) Why is - 2 G x G 2 not a suitable domain for h(x)?
[1]
(ii) Given that h (1) = 4 and h l (1) = 16 , find the value of a and of b.
[2]
MINDBRIDGE
12
f (x) = x 2 + 2x - 3 for x H- 1
(a) Given that the minimum value of x 2 + 2x - 3 occurs when x =- 1, explain why f (x) has an inverse.
[1]
(b) On the axes below, sketch the graph of y = f (x) and the graph of y = f - 1 (x) . Label each graph
and state the intercepts on the coordinate axes.
y
O
x
[4]
MINDBRIDGE
13 The function f is defined by
f (x) = 2 - x + 5
for - 5 G x 1 0 .
(i) Write down the range of f.
[2]
(ii) Find f -1 (x) and state its domain and range.
[4]
The function g is defined by
(iii) Solve fg (x) = 0 .
g (x) =
4
x
for - 5 G x 1 - 1 .
[3]
MINDBRIDGE
f | x 7 e 3x for x d R
g | x 7 2x 2 + 1 for x H 0
14
(i)
Write down the range of g.
[1]
(ii)
Show that f - 1 g ( 62) = ln 5 .
[3]
(iii)
Solve f l (x) = 6gll (x) , giving your answer in the form ln a, where a is an integer.
[3]
MINDBRIDGE
(iv)
On the axes below, sketch the graph of y = g and the graph of y = g -1 , showing the points where the
graphs meet the coordinate axes.
y
O
x
[3]
MINDBRIDGE
15
f (x) = 3 + e x
for x ! R
g (x) = 9x - 5 for x ! R
(a) Find the range of f and of g.
[2]
(b) Find the exact solution of f -1 (x) = g l (x) .
[3]
(c) Find the solution of g 2 (x) = 112 .
[2]
MINDBRIDGE
16
f (x) = 5 + sin
g (x) = x -
r
3
x
for 0 G x G 2r radians
4
for x ! R
(i)
Write down the range of f (x) .
[2]
(ii)
Find f - 1 (x) and write down its range.
[3]
(iii)
Solve 2fg (x) = 11.
[4]
MINDBRIDGE
17 (a)
(b)
f (x) = 4 ln (2x - 1)
(i) Write down the largest possible domain for the function f.
[1]
(ii) Find f - 1 (x) and its domain.
[3]
g (x) = x + 5 for x ! R
h (x) = 2x - 3 for x H
Solve gh (x) = 7 .
3
2
[3]