Inverse and composite functions (Higher tier)

Function notation

f(x) is the output when input x is fed into f. f: x → 3x + 2 is the same as f(x) = 3x + 2.

Composite functions

fg(x) means "apply g first, then apply f to the result". Read right to left.

Example: f(x) = 2x + 1, g(x) = x². Then:

• fg(x) = f(x²) = 2x² + 1.
• gf(x) = g(2x + 1) = (2x + 1)².

In general fg(x) ≠ gf(x).

Inverse functions

f⁻¹(x) "undoes" f. f⁻¹(f(x)) = x.

To find f⁻¹(x):

1. Replace f(x) with y.
2. Swap x and y.
3. Solve for y.
4. That y is f⁻¹(x).

Example: f(x) = 3x + 2. y = 3x + 2 → x = 3y + 2 → y = (x − 2)/3. So f⁻¹(x) = (x − 2)/3.

Useful identities

• ff⁻¹(x) = x.
• f⁻¹f(x) = x.
• (fg)⁻¹(x) = g⁻¹f⁻¹(x). Inverse of a composition reverses the order.

Self-inverse functions

A function with f⁻¹ = f. Common examples:

• f(x) = a − x (e.g. f(x) = 5 − x).
• f(x) = a/x (for any nonzero a).
• f(x) = x.

Restrictions on domain

For inverses to exist, f must be one-to-one (no two inputs share an output). Quadratics fail this on their full domain — restrict (e.g. x ≥ 0) before finding inverse.

Common CCEA exam tip

When asked to "find ff(x)" or "find ff(3)" — substitute carefully. Bracket the inner function value when feeding into the outer to avoid sign errors.