GCSE & IGCSE maths practice

Functions (composite and inverse) questions and answers

A function takes an input and gives an output: if $f(x) = 3x + 2$ then $f(4) = 14$ . A composite function $fg(x)$ means do $g$ first, then $f$ . The inverse function $f^{-1}(x)$ undoes $f$ : write $y = f(x)$ , swap $x$ and $y$ , and rearrange. Type answers like (x+5)/2.

8 questions · Higher tier · Every answer is checked as you type it, with a full worked solution. Updated 27 Oct 2026.

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Where functions (composite and inverse) comes up

GCSE (AQA, Edexcel, OCR)	Higher only: f(x), fg(x) and f⁻¹(x) notation, composite functions and inverse functions.
Cambridge IGCSE (0580)	Extended only: understand functions, domain and range, and use function notation, including inverse and composite functions.
Edexcel International GCSE (4MA1)	Higher only: function notation, domain and range, composite functions fg and inverse functions f⁻¹.

The method

f(a): replace every x with a.
fg(x): put g(x) into f in place of x.
Inverse: y = f(x), swap x and y, make y the subject.
Check: f(f⁻¹(x)) should give x.

Functions (composite and inverse) questions

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Warm-up

Question 1Warm-upHigher

$f(x) = 3x + 2$ . Find $f(4)$ .

Question 2Warm-upHigher

$f(x) = x^2 - 1$ . Find $f(-3)$ .

Exam-style

Question 3Exam-styleHigher

$f(x) = 2x - 5$ . Solve $f(x) = 11$ .

x =

Question 4Exam-styleHigher

$f(x) = 2x - 5$ . Find $f^{-1}(x)$ .

f⁻¹(x) =

Question 5Exam-styleHigher

$f(x) = 3x + 1$ and $g(x) = x^2$ . Find $fg(2)$ .

Question 6Exam-styleHigher

$f(x) = 3x + 1$ and $g(x) = x - 2$ . Find $fg(x)$ in its simplest form.

fg(x) =

Challenge

Question 7ChallengeHigher

$f(x) = \dfrac{x + 4}{3}$ . Find $f^{-1}(x)$ .

f⁻¹(x) =

Question 8ChallengeHigher

$f(x) = \dfrac{1}{x - 2}$ . Which value of $x$ must be excluded from the domain?

x =

Common mistakes

Doing f first in fg(x) (g goes first).
Writing f⁻¹(x) as 1 ÷ f(x).
Squaring before substituting a negative: put it in brackets.

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Sources

The questions are our own. The “where it comes up” notes are from each board's published specification, checked on 26 October 2026:

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