Functions
Function notation, composite and inverse functions.
Review these first
Learning objectives
What you need to be able to do
Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.
- 2.8.1Use function notation and find composite and inverse functions.Supplement
5 minute read
Functions
Notation
f(x) = 3x + 1 is a rule. f(2) means substitute 2 for x, giving 7. The letter in the bracket is only a placeholder.
Composite functions
fg(x) means "do g first, then f" — the function nearer x acts first. With f(x) = 2x + 1 and g(x) = x²:
fg(x) = f(x²) = 2x² + 1gf(x) = g(2x + 1) = (2x + 1)²
These differ, so order matters.
Inverse functions
f⁻¹(x) reverses f. To find it:
- Write
y = f(x). - Swap x and y.
- Rearrange to make y the subject.
For f(x) = 3x − 4: y = 3x − 4 → x = 3y − 4 → y = (x + 4)/3.
Checking
ff⁻¹(x) = x always. Substituting your inverse back is the quickest way to confirm it, and takes one line.
Think of it like this
A function is a machine and its inverse is the same machine run backwards. Composite functions are two machines in a line — and reversing the order of the machines generally changes what comes out.
Worked examples
Method, step by step
Given f(x) = 2x − 3 and g(x) = x + 5, find gf(x) and f⁻¹(x).
- 1gf(x) means f acts first: f(x) = 2x − 3.
- 2Then apply g: g(2x − 3) = (2x − 3) + 5 = 2x + 2.
- 3For the inverse, write y = 2x − 3 and swap: x = 2y − 3.
- 4Rearrange: 2y = x + 3, so y = (x + 3)/2.
gf(x) = 2x + 2 and f⁻¹(x) = (x + 3)/2
Common misconceptions
- Reading `fg(x)` as "f first". The inner function acts first.
- Treating `f⁻¹(x)` as `1/f(x)`. The −1 denotes the inverse, not a reciprocal.
- Forgetting to swap x and y when finding an inverse, which produces the original function rearranged rather than its inverse.
In the exam
- Write out the substitution explicitly, e.g. `fg(x) = f(x²)`, before simplifying — it earns method marks and prevents order errors.
- Verify any inverse with `ff⁻¹(x)`; if it does not simplify to x, something is wrong.