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.
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).
- 1Write y = 2x + 1.
- 2Swap x and y: x = 2y + 1.
- 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
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.
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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).
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