Composite functions
Objective. Apply functions in sequence and track domains.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A composite function applies one function to the output of another, written fg(x) meaning first apply g, then apply f to that result. To evaluate fg(x), first compute g(x), then substitute that expression into f. For f(x) = x + 3 and g(x) = 2x, fg(x) = f(g(x)) = f(2x) = 2x + 3, while gf(x) = g(x+3) = 2(x+3) = 2x + 6, showing order matters. When finding fg(5), it is often easiest to find g(5) first, then feed that number into f. Always keep track of the domain, since some inputs may be invalid for the inner function.
Worked example
Given f(x) = x − 2 and g(x) = x², find fg(3).
- 1Apply the inner function first: g(3) = 3² = 9
- 2Substitute this result into f: f(9) = 9 − 2
- 3Simplify: f(9) = 7
- 4So fg(3) = 7
Answer: 7
Concept mastery
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Practice
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Start practisingCommon mistake. Pupils work out fg(x) by applying f first instead of g first, reversing the intended order. For f(x)=x+1, g(x)=3x, they might compute fg(x) as 3(x+1) instead of the correct 3x+1. Always apply the function written nearest to x (the inner one) first.
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