CAIE A-Level Mathematics 1.2 Functions Question Bank

CAIE A-Level Mathematics 1.2 Functions Question Bank
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise domains and ranges, composite and inverse functions and graph transformations while checking restrictions, operation order and exact function notation.

Exam points

  • state domains and ranges with the correct variable, inequality and included endpoint
  • form fg or gf in the stated order and verify that the inner range fits the outer domain
  • find an inverse by restricting one-to-one branches and solve transformation or composite equations

Covered syllabus points

1.2.1—Function terminology1.2.2—Range and composition1.2.3—One-one and inverse functions1.2.4—One-one and inverse functions1.2.5—Graph transformations

Question 3

Question 3(a)

(a)

The graph of y=f(x) is transformed to the graph of y=f(3 x)+2.
Describe fully the two transformations which have been combined to give the resulting graph.

[ 3 ]

Question 3(b)

(b)

A different graph has equation y=g(x). This graph is stretched by scale factor 3 in the y-direction and then reflected in the y-axis.

Write down the equation of the transformed graph in terms of the function g.

[ 2 ]

Question 6

[Maximum number: 7]

Functions f and g are defined by

f(x)=(x+3)2−12 for x⩾0, g(x)=2x−5 for x∈R.\begin{array}{ll} \mathrm{f}(x)=(x+3)^{2}-12 & \text { for } x \geqslant 0, \\ \mathrm{~g}(x)=2 x-5 & \text { for } x \in \mathbb{R} . \end{array}

Question 6(a)

(a)

State the range of f.

[ 1 ]

Question 6(b)

(b)

Find an expression for f−1(x)\mathrm{f}^{-1}(x).

[ 2 ]

Question 6(c)

(c)

Solve the equation gf(x)=69.

[ 4 ]

Question 8

[Maximum number: 8]

Functions f and g are defined by

f(x)=(x+a)2−a for x⩽−a, g(x)=2x−1 for x∈R,\begin{aligned} & \mathrm{f}(x)=(x+a)^{2}-a \text { for } x \leqslant-a, \\ & \mathrm{~g}(x)=2 x-1 \text { for } x \in \mathbb{R}, \end{aligned}

where a is a positive constant.

Question 8(a)

(a)

Find an expression for f−1(x)\mathrm{f}^{-1}(x).

[ 3 ]

Question 8(b)(i)

(b)

State the domain of the function f−1\mathrm{f}^{-1}.

[ 1 ]

Question 8(b)(ii)

(c)

State the range of the function f−1\mathrm{f}^{-1}.

[ 1 ]

Question 8(c)

(d)

Given that a=72a=\frac{7}{2}, solve the equation gf(x)=0.

[ 3 ]

Question 10

[Maximum number: 10]

The functions f and g are defined by
f(x)=xfor x⩾0,f(x)=\sqrt{x}\quad\text{for }x\geqslant 0,
and
g(x)=3x+2−5for x⩾−2.g(x)=3\sqrt{x+2}-5\quad\text{for }x\geqslant -2.

Question 10(a)

(a)

Describe fully a sequence of transformations which transforms the graph of y=f(x) to the graph of y=g(x). You should make clear the order in which the transformations are applied.

[ 5 ]

Question 10(b)

(b)
Figure for Question 10(b) — CAIE A-Level Mathematics

The diagram shows the graph of y=g(x).

On the diagram sketch the graph of y=g−1(x)y=g^{-1}(x) together with any relevant mirror line.

[ 2 ]

Question 10(c)

(c)

Find an expression for g−1(x)\mathrm{g}^{-1}(x).

[ 2 ]

Question 10(f)

(d)

Explain why the composite function hg−1\mathrm{hg}^{-1} cannot be formed.

[ 1 ]

Question 10

[Maximum number: 7]

The function f is defined by

f(x)=3+7x−2\mathrm{f}(x)=3+\frac{7}{x-2}

for x>2.

Question 10(b)

(a)

Find an expression for f−1(x)\mathrm{f}^{-1}(x) and state the domain of f−1\mathrm{f}^{-1}.

[ 4 ]

Question 10(c)

(b)

The function g is defined by

g(x)=1+4x2x−3g(x)=\frac{1+4 x}{2 x-3}

for x>32x>\frac{3}{2}.
Show that fg⁡(x)≡kx\operatorname{fg}(x) \equiv k x, where k is a constant to be determined.

[ 3 ]