Key Terms
- Composite function: A function applied to the result of another function
- fg(x): Apply g first, then apply f to the result — means f(g(x))
- gf(x): Apply f first, then apply g to the result — means g(f(x))
- Order matters: fg(x) and gf(x) usually give different answers
- f²(x): Apply f twice — means f(f(x))
Must-Know Facts
- fg(x) means do g first, then do f — read right to left
- fg(x) ≠ gf(x) in general — order matters
- fg(x) ≠ f(x) × g(x) — it is substitution, not multiplication
- To find fg(3): first calculate g(3), then substitute that answer into f
- f⁻¹f(x) = x — inverse and function together always give x back
Key Methods
- Evaluate fg(a): calculate g(a) first, then apply f to that result
- Find fg(x) as an expression: substitute g(x) into f in place of x
- Simplify: expand and collect like terms after substitution
- Solve fg(x) = k: build the composite, set equal to k, solve
Common Mistakes
- Applying functions in the wrong order: fg(x) means apply g FIRST then f — not f first; always read the composite right to left
- Treating fg(x) as multiplication: fg(x) is NOT f(x) × g(x) — it is substitution of g(x) into f
- Errors in algebraic substitution: When finding fg(x) as an expression, substitute the full expression for g(x) into f and use brackets — e.g. if g(x) = 2x + 1, replace every x in f with (2x + 1)
- Confusing fg and gf: fg(3) and gf(3) usually give different answers — always identify which function is applied first
This topic summary covers Knowledge Organiser: Composite Functions within Composite Functions for GCSE Mathematics. Revise Composite Functions in Algebra for GCSE Mathematics with 14 exam-style questions and 12 flashcards. This is a high-frequency topic, so it is worth revising until the explanation feels precise and repeatable. It is section 8 of 8 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Practice questions for Composite Functions
The composite function fg(x) means: A) Apply g first, then apply f to the result B) Apply f first, then apply g to the result C) Multiply f(x) by g(x) D) Add f(x) to g(x)
Explain why, in general, fg(x) and gf(x) are NOT equal. You may use an example to support your explanation.