$\mathrm{f}(x)=2x^2-3$. Find $\mathrm{f}(-2)$ and solve $\mathrm{f}(x)=15$.

$\mathrm{f}(-2)=2\times4-3=5$. $2x^2-3=15$: $x^2=9$, $x=\pm3$. Answer: $\mathrm{f}(-2)=5$; $x=\pm3$

(a) Which value is excluded from the domain of $\mathrm{f}(x)=\dfrac{1}{x-2}$? (b) State the range of $\mathrm{g}(x)=x^2+3$.

(a) The denominator cannot be 0: exclude $x=2$. (b) $x^2\ge0$, so $\mathrm{g}(x)\ge3$. Answer: (a) $x=2$ (b) $\mathrm{g}(x)\ge3$

$\mathrm{f}(x)=x+4$ and $\mathrm{g}(x)=3x^2$. Find $\mathrm{fg}(x)$ and $\mathrm{fg}(2)$.

$\mathrm{fg}(x)$ means apply g first: $\mathrm{f}(3x^2)=3x^2+4$. $\mathrm{fg}(2)=3\times4+4=16$. Answer: $3x^2+4$; 16

$\mathrm{f}(x)=\dfrac{2x+1}{5}$. Find $\mathrm{f}^{-1}(x)$.

Write $y=\dfrac{2x+1}{5}$ and swap: $x=\dfrac{2y+1}{5}$. $5x=2y+1$, so $y=\dfrac{5x-1}{2}$. Answer: $\mathrm{f}^{-1}(x)=\dfrac{5x-1}{2}$

$ \mathrm{g}(x) = \dfrac{1}{\sqrt{x - 5}} $. Find the largest possible domain of g.

M1 x − 5 > 0 (the square root must be positive: it cannot be negative, and 0 would mean dividing by 0) A1 x > 5 Answer: $ x \gt 5 $

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Edexcel IGCSE Maths (4MA1) · Unit 4: Graphs & Functions

Edexcel IGCSE 4MA1 Functions: Exam Questions and Answers

Functions questions with full solutions for Pearson Edexcel International GCSE Mathematics A (4MA1), Higher tier. Read the 4 worked examples first, then try the 8 exam-style questions yourself and check your working against the mark scheme.

4 worked examples
8 questions with mark schemes
Higher tier · calculator allowed
Free, no sign-in

Practise functions in Unit 4 →Graphs, Sequences and Calculus revision notes

What this topic covers

Functions 1: Function Notation: Reading f(x) notation.
Functions 2: Domain and Range: Stating domain / range from a graph.
Functions 3: Composite Functions fg(x): Composite functions fg(x) and gf(x).
Functions 4: Inverse Functions f⁻¹(x): Finding inverse functions.

Functions worked examples

Worked example 1: Function notation

$f (x) = 2 x^{2} - 3$ . Find $f (- 2)$ and solve $f (x) = 15$ .

Solution

$f (- 2) = 2 \times 4 - 3 = 5$ .
$2 x^{2} - 3 = 15$ : $x^{2} = 9$ , $x = \pm 3$ .

3 steps

Tip: f(−2) means replace every x with −2.

Worked example 2: Domain and range

(a) Which value is excluded from the domain of $f (x) = \frac{1}{x - 2}$ ? (b) State the range of $g (x) = x^{2} + 3$ .

Solution

(a) The denominator cannot be 0: exclude $x = 2$ .
(b) $x^{2} \geq 0$ , so $g (x) \geq 3$ .

3 steps

Tip: Domain is the inputs; range is the outputs.

Worked example 3: Composite functions

$f (x) = x + 4$ and $g (x) = 3 x^{2}$ . Find $fg (x)$ and $fg (2)$ .

Solution

$fg (x)$ means apply g first: $f (3 x^{2}) = 3 x^{2} + 4$ .
$fg (2) = 3 \times 4 + 4 = 16$ .

3 steps

Tip: fg(x) ≠ gf(x) in general: gf(x) = 3(x + 4)².

Worked example 4: Inverse function

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

Solution

Write $y = \frac{2 x + 1}{5}$ and swap: $x = \frac{2 y + 1}{5}$ .
$5 x = 2 y + 1$ , so $y = \frac{5 x - 1}{2}$ .

3 steps

Tip: Check: f⁻¹(f(3)) = f⁻¹(7/5) = 3.

Where students lose marks on functions

Examiner's noteInverse functions: collecting x terms

Where marks are lost: The method is known but the algebra of collecting terms and factorising x out breaks down.

Examiners flagged this in 2 exam series (November 2023, June 2024). Summarised in our own words from Pearson Edexcel examiner reports. Read the full note →

Functions exam-style questions

Try each question before you open the answer (one has its full worked solution open). The mark schemes use Edexcel-style marks: M for method, A for accuracy (after the method mark), B for an independent result.

Question 1 · easy · 3 marks

$f (x) = 4 - 3 x$ . (a) Work out $f (- 2)$ . (b) Find the value of $x$ for which $f (x) = 19$ .

Show the answer and mark scheme

B1 10
M1 4 − 3x = 19
A1 −5

Answer: (a) 10 (b) $x = - 5$

Question 2 · easy · 1 mark

State the value of $x$ that must be excluded from the domain of $f (x) = \frac{5}{x + 4}$ .

Show the answer and mark scheme

B1 −4

Answer: $x = - 4$

Question 3 · easy · 2 marks

$f (x) = 3 x - 2$ with domain $1 \leq x \leq 6$ . Find the range of f.

Show the answer and mark scheme

M1 works out f(1) = 1 or f(6) = 16
A1 1 ≤ f(x) ≤ 16

Answer: $1 \leq f (x) \leq 16$

Question 4 · medium · 3 marks

$g (x) = x^{2} - 2 x$ . Solve $g (x) = 15$ .

Show the answer and mark scheme

M1 x² − 2x − 15 = 0
M1 (x − 5)(x + 3) = 0
A1 x = 5, x = −3

Answer: $x = 5$ or $x = - 3$

Question 5 · medium · 2 marks

$g (x) = \frac{1}{\sqrt{x - 5}}$ . Find the largest possible domain of g.

Full worked solution Free, no sign-in

M1 x − 5 > 0 (the square root must be positive: it cannot be negative, and 0 would mean dividing by 0)
A1 x > 5

Answer: $x > 5$

Every question here has a full worked solution. See them all →

Question 6 · hard · 4 marks

$f (x) = a x + b$ , where $a$ and $b$ are constants. $f (2) = 11$ and $f (- 3) = - 9$ . (a) Find the values of $a$ and $b$ . (b) Solve $f (2 x) = f (x) + 20$ .

Show the answer and mark scheme

M1 2a + b = 11 and −3a + b = −9
A1 a = 4, b = 3
M1 8x + 3 = 4x + 3 + 20
A1 x = 5

Answer: (a) $a = 4$ , $b = 3$ (b) $x = 5$

Question 7 · hard · 5 marks

A stone is thrown vertically upwards from ground level. Its height, $h$ metres, $t$ seconds after it is thrown is given by $h (t) = 40 t - 5 t^{2}$ . (a) Work out $h (2)$ . (b) Solve $h (t) = 35$ . (c) Show that $h (4 + p) = h (4 - p)$ for every value of $p$ .

Show the answer and mark scheme

B1 60
M1 5t² − 40t + 35 = 0 or t² − 8t + 7 = 0
A1 t = 1, t = 7
M1 40(4 + p) − 5(4 + p)² expanded: 160 + 40p − 80 − 40p − 5p²
A1 both simplify to 80 − 5p²

Answer: (a) 60 m (b) $t = 1$ or $t = 7$ (c) both equal $80 - 5 p^{2}$ (shown)

Question 8 · hard · 3 marks

$f (x) = x^{2} + 8 x + 19$ , where $x$ can be any real number. By completing the square, find the range of f.

Show the answer and mark scheme

M1 (x + 4)² …
A1 (x + 4)² + 3
A1 f(x) ≥ 3

Answer: $f (x) \geq 3$

Every question and worked example here is original, written for IGCSE Math Revision rather than copied from Pearson papers, and each answer was re-solved independently before publishing.

Revise functions for the exam

Unit 4 practice: Graphs & Functions in the student hub (more questions with mark schemes)
IGCSE Maths revision notes for this topic and the IGCSE Maths formula sheet
Edexcel IGCSE Maths 4MA1 past papers to practise under exam conditions
IGCSE Maths 4MA1 examiner report summary: where students lose marks

Questions students ask

Which Edexcel IGCSE Maths papers test functions?

Either Higher tier paper, 1H or 2H, can test functions: both cover the whole Pearson Edexcel International GCSE Mathematics A (4MA1) specification and both allow a calculator. Show your method, because most marks are for working.

Where can I practise more functions questions?

In the IGCSE student hub: Unit 4 (Graphs & Functions) has more exam-style questions on this topic, each with a mark scheme. Practice starts with a 7-day free trial; the worked examples and questions on this page are free.

Practise functions with your progress saved

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Edexcel IGCSE 4MA1 Functions: Exam Questions and Answers

Try it first

What this topic covers

Functions worked examples

Worked example 1: Function notation

Solution

Worked example 2: Domain and range

Solution

Worked example 3: Composite functions

Solution

Worked example 4: Inverse function

Solution

Where students lose marks on functions

Functions exam-style questions

Question 1 · easy · 3 marks

Question 2 · easy · 1 mark

Question 3 · easy · 2 marks

Question 4 · medium · 3 marks

Question 5 · medium · 2 marks

Full worked solution Free, no sign-in

Question 6 · hard · 4 marks

Question 7 · hard · 5 marks

Question 8 · hard · 3 marks

Revise functions for the exam

Questions students ask

Practise functions with your progress saved

More Unit 4 topics: Graphs & Functions