The idea

Composite and inverse functions

How composite functions chain together (and why order matters), and how to find an inverse — what it does, how to build it, and the link to its graph.

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Pure · Algebra & functions

Composite and inverse functions

How composite functions chain together (and why order matters), and how to find an inverse — what it does, how to build it, and the link to its graph.

Why it works

Chaining rules: composites

Functions questions at A-level live and die on two instincts: read composites inside-out, and treat

f^{-1}

as undo, never as a reciprocal. A function is a rule with an input and an output, plus a domain (allowed inputs) and range (resulting outputs).

Composite functions chain two rules.

fg(x)

means do $g$ first, then $f$ — read it inside-out, like nested brackets:

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

. So if

f(x) = 2x + 1

and

g(x) = x^2

, then

fg(x) = f(x^2) = 2x^2 + 1

, whereas

gf(x) = g(2x + 1) = (2x + 1)^2

. Order matters —

fg \ne gf

in general.

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