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 functions $f$ and $g$ are defined by $f(x) = x + 2$ and $g(x) = x^{2}$ .

Solve the equation $gf(x) = 25$ . (4 marks)
Level 3 · 4 marksTry it
2
The function $\mathrm{f}$ is defined by

$\mathrm{f}(x) = x^2 - 4x + 7, \qquad x \geqslant 2$
(a) Express $\mathrm{f}(x)$ in the form $(x - a)^2 + b$ , where $a$ and $b$ are constants. (2 marks)
(b) State the range of $\mathrm{f}$ . (1 mark)
(c) Find $\mathrm{f}^{-1}(x)$ and state its domain. (3 marks)
Level 3 · 6 marksTry it
3
The functions $f$ and $g$ are defined by $f(x) = 2x + 1$ and $g(x) = x^2$ . Solve $gf(x) = 9$ .
Level 3 · 3 marksTry it
4
The function $f$ is defined by $f(x) = 3x - 2$ . Find $f^{-1}(x)$ .
Level 3 · 2 marksTry it
5
The function $f$ is defined by

$f(x) = 9 - 2\sqrt{x + 4} \qquad x \in \mathbb{R},\ x \geqslant -4$
(a) State the range of $f$ .

(1 mark)
(b) Find $f^{-1}(x)$ .

(3 marks)
Level 3 · 4 marksTry it

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