MathsGradeUp logoMathsGradeUp
MathematicsInternational GCSEEdexcel & Cambridge (CIE)Years 10–11 Calculator & non-calculator

IGCSE Maths Functions (f(x)) Notes

A function is a rule that turns an input into a single output. IGCSE tests function notation, composite functions and inverse functions.

Practise this topic free Ask Study Coach

Key concepts

Function notation
f(x) is the output of function f for input x; substitute the value in place of x.
Composite function
fg(x) means do g first, then apply f to the result — work from the inside out.
Inverse function
f⁻¹(x) reverses f. Swap x and y, then rearrange to make y the subject.

Key formulas

Composite order

fg(x) = f(g(x))

Apply the closest function to x first.

Worked example

f(x) = 2x + 1. Find f⁻¹(x).

  1. 1Write y = 2x + 1.
  2. 2Swap x and y: x = 2y + 1.
  3. 3Rearrange for y: y = (x − 1)/2.

Answer: f⁻¹(x) = (x − 1)/2

Common mistakes

✗ Doing fg(x) in the wrong order.

✓ fg(x) means apply g first, then f — inside out.

✗ Forgetting to swap x and y for the inverse.

✓ Swap the variables first, then rearrange to make y the subject.

✗ Treating f(x) as f × x.

✓ It is a function output — substitute the input for x.

Quick check

Try these, then reveal the answers.

f(x) = 3x − 2, find f(5).Reveal answer

13

f(x) = x + 1, g(x) = 2x. Find fg(3).Reveal answer

7

f(x) = x − 4, find f⁻¹(x).Reveal answer

x + 4

Practise more questions with Study Coach feedback

Exam tips

  • Write out g(x) fully before substituting it into f for composite functions.
  • Check an inverse by confirming f(f⁻¹(x)) returns x.
  • State the order clearly — fg(x) and gf(x) are usually different.

Related practice

Practise this topic

Exam-style questions with instant Study Coach feedback.

Flashcards

Quick-fire recall to lock in the key facts.

Worksheets

Printable practice you can work through offline.

Past papers

Full papers with examiner-grade marking.

Study Coach

Ask for a step-by-step explanation any time.

All Mathematics

Every Mathematics topic across every exam board.

Related topics

AlgebraCalculus

Key terms explained

Short, exam-ready definitions with a formula, a worked example and an examiner tip.

What is area under a curve?What is chain rule?What is completing the square?What is definite integral?What is differentiation?What is discriminant?Full glossary

Frequently asked questions

What is a composite function?+

A function applied to the output of another, written fg(x) = f(g(x)); do the inner function first.

How do I find an inverse function?+

Write y = f(x), swap x and y, then rearrange to make y the subject — that is f⁻¹(x).

Create a free account to practise functions (f(x)) with Study Coach feedback

Try 5 questions now, get instant examiner-grade marking and ask the Study Coach to explain anything. Free during our early launch period — no credit card required.

Create free account Try the Study Coach

Related on MathsGradeUp

Keep studying

Practise it