Composite functions

Objective. Compose functions in the correct order and track restrictions.

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2. Worked example
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Learn: the key idea

A composite function fg(x) means apply g first, then apply f to the result, so fg(x)=f(g(x)), and the order matters because fg and gf are usually different functions. The domain of fg(x) must respect both the domain of g and the domain of f applied to g's output, so any value excluded at either stage is excluded overall. For f(x)=2x+1 and g(x)=x², fg(x)=f(g(x))=2x²+1, while gf(x)=g(f(x))=(2x+1)²; these are clearly different expressions. Always substitute the inner function completely before simplifying the outer one.

Worked example

Given f(x)=3x−2 and g(x)=x+4, find the composite function fg(x).

1 $Identify that fg(x) means f(g(x)), so apply g first$
2 $Substitute g(x)=x+4 into f: f(x+4)=3(x+4)-2$
3 $Expand the bracket: 3x+12-2$
4 $Simplify to get fg(x)=3x+10$

Answer: fg(x) = 3x + 10

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Common mistake. Students compute gf(x) when asked for fg(x), reversing the order; for f(x)=x+3 and g(x)=2x, fg(x)=2x+3 but a student writes x+6, having applied f first by mistake.

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