Sheet № 96 · Higher only · AQA · Edexcel · OCR
Composite Functions –
Composite functions combine two or more functions by applying them one after the other. They appear regularly on Higher-tier GCSE papers and are worth understanding well because the method is systematic once you grasp the order of operations.
§Key definitions
Question:
f(x) = 3x + 1 and g(x) = x². Find fg(2).
Answer:
fg(2) = 13
(a)
fg(x) = f(g(x)) = f(x² + 1)
(b)
gf(x) = g(f(x)) = g(2x − 5)
Q1 (Higher):
f(x) = 5x − 2 and g(x) = x + 4. Find fg(3).
§Formulas to memorise
fg(x) = f(g(x)) — apply g first, then f
gf(x) = g(f(x)) — apply f first, then g
ff(x) = f(f(x)) — apply f twice
Identify — which function is applied first (the inner function, closest to x).
Write down — the expression for the inner function.
Substitute — that entire expression into the outer function, replacing every x.
Simplify — the result.
Worked example
f(x) = 3x + 1 and g(x) = x². Find fg(2).
Working:
⚠ Common mistakes
- ✗Applying functions in the wrong order. fg(x) means g first, then f. The function written first (f) is applied second. Always work from the inside out.
- ✗Forgetting to replace every x. When substituting g(x) into f(x), every x in the expression for f must be replaced with the entire expression for g(x), not just part of it.
- ✗Not expanding brackets fully. When gf(x) involves squaring a linear expression, you must expand (2x − 5)² correctly as 4x² − 20x + 25, not 4x² + 25.
✦ Exam tips
- →To evaluate fg(2), you can either find the composite expression fg(x) first and substitute 2, or find g(2) and then apply f to that number. Both are valid; the second is often quicker.
- →Show intermediate substitution clearly — for example, write "f(g(x)) = f(x² + 1) = 2(x² + 1) − 5" rather than jumping to the answer.
- →If asked to solve fg(x) = k, find the composite expression first, then solve the resulting equation.