Functions, inverses and composites
Functions appear on WJEC Higher Unit 1 and Unit 2. The notation looks intimidating but the rules are mechanical.
Function notation
f(x) = 3x − 2 is a rule: input x, multiply by 3, subtract 2. To evaluate f(5): replace x with 5 → f(5) = 3(5) − 2 = 13.
You may see g(x), h(x) used alongside f(x) in the same question — each is a separate rule.
Inverse functions
The inverse f⁻¹(x) reverses f. Method:
- Write y = f(x).
- Swap x and y.
- Solve for y. That expression is f⁻¹(x).
Example: f(x) = 3x − 2.
- y = 3x − 2
- Swap: x = 3y − 2
- Solve: y = (x + 2) / 3
- f⁻¹(x) = (x + 2) / 3.
Check: f⁻¹(13) = (13 + 2) / 3 = 5 ✓ (the original input).
Composite functions
fg(x) means apply g first, then f. So fg(x) = f(g(x)). Notation order is RIGHT-TO-LEFT.
Example: f(x) = 3x − 2, g(x) = x + 4.
- fg(x) = f(x + 4) = 3(x + 4) − 2 = 3x + 12 − 2 = 3x + 10.
- gf(x) = g(3x − 2) = (3x − 2) + 4 = 3x + 2.
Notice fg ≠ gf in general.
Domain and range (informal)
WJEC may ask for the domain (set of allowed inputs) and range (set of possible outputs). For f(x) = √x the domain is x ≥ 0 since you cannot square-root a negative number.
WJEC exam tip
For composites, write the working line by line: "g(x) = …, then f(g(x)) = …". The first substitution earns M1, the simplification earns A1. Don't skip steps — examiners will not award the A1 if you only show the final answer.
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