Composite functions

Composition substitutes an entire output into the next function. Brackets keep the substituted expression intact. In general fg and gf differ; composition is not ordinary multiplication.

The domain requires x to belong to g’s domain and g(x) to belong to f’s domain. A simplified formula may conceal one of these requirements. Track the restrictions through the two operations before simplifying.

Worked example

Given f(x) = x² + 1 and g(x) = 2x − 3, find fg(x).

1. Start with g(x) = 2x − 3.
2. Substitute into f: f(g(x)) = (2x − 3)² + 1.
3. Expand 4x² − 12x + 9 + 1.

Answer: 4x² − 12x + 10

Revise first: Expanding and factorising.