Composite & Inverse Functions

This lesson covers composite functions and inverse functions as required by the AQA A-Level Mathematics specification (7357). Composing functions means applying one function after another, while an inverse function reverses the effect of the original function. These concepts are fundamental to A-Level mathematics and appear frequently in exam papers.

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Composite Functions

A composite function is formed by applying one function and then another. The notation fg(x) or f ∘ g(x) means "apply g first, then apply f to the result".

fg(x) = f(g(x))

Important: fg(x) means apply g first, then f. The order matters — in general, fg(x) ≠ gf(x).

Evaluating Composite Functions

Example: Given f(x) = 2x + 3 and g(x) = x², find:

(a) fg(2):

g(2) = 4
fg(2) = f(4) = 2(4) + 3 = 11

(b) gf(2):

f(2) = 7
gf(2) = g(7) = 49

(c) fg(x):

fg(x) = f(g(x)) = f(x²) = 2x² + 3

(d) gf(x):

gf(x) = g(f(x)) = g(2x + 3) = (2x + 3)²