Functions, inverses and composites
A function takes an input and produces a unique output. OCR J560 introduces function notation on Higher (J560/04–06) and tests inverses + composites with full algebraic rigour.
Function notation
f(x) means "the output of function f when the input is x".
Example: f(x) = 2x + 5.
- f(3) = 2(3) + 5 = 11.
- f(−1) = 2(−1) + 5 = 3.
The variable can be anything — f(a), f(2x), f(x + 1) all mean "substitute that into the rule".
Inverse functions
The inverse f⁻¹(x) "undoes" f. If f(a) = b, then f⁻¹(b) = a.
To find f⁻¹:
- Write y = f(x).
- Swap x and y: x = f(y).
- Solve for y.
- Replace y with f⁻¹(x).
Example: f(x) = 3x − 4.
- y = 3x − 4 → swap: x = 3y − 4 → solve: y = (x + 4)/3.
- f⁻¹(x) = (x + 4)/3.
Check: f(2) = 2; f⁻¹(2) = (2 + 4)/3 = 2. ✓
Composite functions
fg(x) means "apply g first, then apply f to the result". The order is right-to-left in the notation.
Example: f(x) = 2x + 1, g(x) = x².
- fg(x) = f(g(x)) = f(x²) = 2x² + 1.
- gf(x) = g(f(x)) = g(2x + 1) = (2x + 1)².
Note: fg(x) ≠ gf(x) in general.
Domain and range (informal at GCSE)
- Domain: set of allowed inputs.
- Range: set of possible outputs.
For f(x) = √x, domain is x ≥ 0; range is f(x) ≥ 0.
OCR mark scheme conventions
- B1 for substituting correctly into f(x).
- M1 for using the right composite order: fg means f(g(x)).
- A1 for the simplified expression or numerical value.
- For inverses: M1 for swap, A1 for the rearranged form.
⚠Common mistakes
- Writing fg(x) but computing gf(x).
- Forgetting brackets when squaring an entire expression in g(f(x)).
- Stopping inverse-finding partway — must isolate y fully.
- Treating f⁻¹ as 1/f (it's NOT a reciprocal).
AI-generated · claude-opus-4-7 · v3-ocr-maths-leaves