Mr. Integral
Math notes/Pre-Calculus/Functions Review

Defining Composition — Cheat sheet

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Key method

(f∘g)(x)=f(g(x)).(f\circ g)(x)=f(g(x)).

Its domain contains exactly the inputs in the domain of gg whose outputs lie in the domain of ff. Both stages must be legal.

Example

Let f(u)=uf(u)=\sqrt u and g(x)=x−3g(x)=x-3. Then (f∘g)(x)=x−3(f\circ g)(x)=\sqrt{x-3} with domain x≥3x\ge3. Reversing the order gives (g∘f)(x)=x−3(g\circ f)(x)=\sqrt x-3 with domain x≥0x\ge0. The different order changes both the rule and its domain.

Avoid this mistake

Composition is not multiplication. f(g(x))f(g(x)) generally differs from f(x)g(x)f(x)g(x).