A-level · Algebra & functions

Composite and inverse functionsExam-style questions

Chaining functions together and running them backwards. Composite functions, read inside-out, and why the order matters; solving an equation with a composite, and the domain check it needs; finding an inverse by rearranging, and why it is not a reciprocal; the inverse's graph as a reflection in y = x; the domain and range swapping over; and why only a one-to-one function has an inverse.

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42 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
The function $f$ is defined by $f(x) = x^2 + 2$ for $x \ge 0$ , and $g(x) = 3x - 1$ .
(a) Find $fg(x)$ .
(b) Hence solve $fg(x) = 11$ .
Level 5 · 5 marksTry it
2
The function $f$ is defined by $f(x) = (x - 1)^{2} + 4$ for $x \geqslant 1$ .
(a) Explain why $f^{-1}$ exists for this domain. (1 mark)
(b) Find $f^{-1}(x)$ , stating its domain. (4 marks)
Level 5 · 5 marksTry it
3
The functions $f$ and $g$ are defined by

$f(x) = \frac{5x + 3}{x + 1} \qquad x \in \mathbb{R},\ x > -1$

$g(x) = 2x - 9 \qquad x \in \mathbb{R}$
(a) Find $fg(6)$ .

(2 marks)
(b) Show that $f(x) = A + \dfrac{B}{x + 1}$ , where $A$ and $B$ are constants to be found.

(2 marks)
(c) Hence state the range of $f$ .

(1 mark)
(d) Find $f^{-1}(x)$ , stating the domain of $f^{-1}$ .

(3 marks)

The function $h$ is defined by

$h(x) = x^2 - 8x + k \qquad x \in \mathbb{R}$

where $k$ is a constant.
(e) Find the range of values of $k$ for which the equation

$g(x) = h(x)$

has no real solutions.

(3 marks)
Level 5 · 11 marksTry it
4
The functions f and g are given by

$f(x) = 3x^2 - 6x \quad$ and $\quad g(x) = 6x - 9$ ,

each defined for all real values of $x$ .
(a) Find the range of f.

(3 marks)
(b) Explain why f does not have an inverse.

(1 mark)
(c) The constant $a$ is such that $fg(1) = g^{-1}(a)$ . Determine the value of $a$ .

(4 marks)
(d) Using set notation, determine the values of $x$ for which $f(x) > g(x)$ .

(3 marks)
Level 5 · 11 marksTry it
5
The function $f$ is defined by $f(x) = \dfrac{3x + 5}{x - 3}$ , $\; x \neq 3$ .

Show that $ff(x) = x$ , and state what this tells you about $f^{-1}$ . (4 marks)
Level 5 · 4 marksTry it

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