Understanding Composite Functions Revision Notes | GCSE AQA Higher Further Maths

1. What Is a Composite Function?

Imagine a vending machine that first doubles your money, then adds 50p. Two separate operations, one after the other — that chain of actions is exactly what a composite function is.

In maths, a composite function is what you get when the output of one function is fed directly into another. You are not doing anything new to the algebra — you are applying functions in a specific sequence.

The key insight is this: the function written closest to x always acts first. In $fg(x)$, that is $g$. In $gf(x)$, that is $f$. The letter on the outside acts last.

2. Evaluating a Composite Function at a Value

Before finding a general expression for $fg(x)$, it helps to practise with a specific number. This makes the two-stage process concrete and straightforward to check.

Suppose $f(x) = 2x + 1$ and $g(x) = x^2$. To find $fg(3)$, work from the inside out: find $g(3)$ first, then apply $f$ to that answer.

Worked example: Evaluating fg(3) Step by Step

Given $f(x) = 2x + 1$ and $g(x) = x^2$, find $fg(3)$.

1. $fg(3)$ means apply $g$ first, then $f$. — g is closest to the input 3, so it acts first. [M1 — correct order identified]
2. $g(3) = 3^2 = 3 \times 3 = 9$ — Substitute $x = 3$ into $g(x) = x^2$. The result is 9.
3. $f(9) = 2(9) + 1 = 18 + 1 = 19$ — Now substitute $x = 9$ into $f(x) = 2x + 1$. [A1 — correct final value]

Answer: $fg(3) = 19$

3. Finding a General Expression for fg(x)

Evaluating at a single number is useful for checking, but exam questions often ask you to find $fg(x)$ as an algebraic expression. The method is identical — substitute $g(x)$ wherever you see $x$ in $f$.

Worked example: Finding fg(x) as an Expression

Given $f(x) = 2x + 1$ and $g(x) = x^2$, find an expression for $fg(x)$.

1. $fg(x) = f(g(x))$ — g acts first, so replace the input of f with the entire expression g(x). [M1]
2. $fg(x) = f(x^2) = 2(x^2) + 1$ — Replace every x in $f(x) = 2x + 1$ with $x^2$.
3. $fg(x) = 2x^2 + 1$ — This is the final simplified expression. [A1]

Answer: $fg(x) = 2x^2 + 1$

Now compare that with $gf(x)$. Here $f$ acts first, so you substitute $f(x)$ into $g$. Watch how the result differs.

Worked example: Finding gf(x) as an Expression

Given $f(x) = 2x + 1$ and $g(x) = x^2$, find an expression for $gf(x)$.

1. $gf(x) = g(f(x))$ — f acts first this time, so replace the input of g with f(x). [M1]
2. $gf(x) = g(2x + 1) = (2x + 1)^2$ — Replace every x in $g(x) = x^2$ with $(2x + 1)$.
3. $(2x + 1)^2 = (2x + 1)(2x + 1) = 4x^2 + 4x + 1$ — Expand carefully: $2x \times 2x = 4x^2$, $2x \times 1 = 2x$, $1 \times 2x = 2x$, $1 \times 1 = 1$. [A1]

Answer: $gf(x) = 4x^2 + 4x + 1$

Notice that $fg(x) = 2x^2 + 1$ and $gf(x) = 4x^2 + 4x + 1$ — these are different expressions. Composite functions are not commutative: the order you apply them in changes the result.

4. Worked Exam-Style Questions

Exam questions on composite functions come in two main forms: evaluate at a number, or find a general expression. Both follow the same two-stage process. Here is a slightly more demanding example to build confidence with less familiar functions.

Worked example: Composite Function with a Rational Function

Given $f(x) = \dfrac{1}{x}$ and $g(x) = 3x - 2$, find $fg(x)$ and evaluate $fg(1)$.

1. $fg(x) = f(g(x)) = f(3x - 2)$ — g acts first, so substitute g(x) = 3x - 2 as the input to f. [M1]
2. $f(3x - 2) = \dfrac{1}{3x - 2}$ — In $f(x) = \frac{1}{x}$, replace x with $(3x - 2)$. [A1]
3. $fg(x) = \dfrac{1}{3x - 2}$ — This is the simplified composite expression.
4. $fg(1) = \dfrac{1}{3(1) - 2} = \dfrac{1}{3 - 2} = \dfrac{1}{1} = 1$ — Substitute x = 1 into the composite expression. [A1]

Answer: $fg(x) = \dfrac{1}{3x - 2}$, and $fg(1) = 1$

5. Practice Problems

The best way to lock in the method is to work through problems yourself before checking the answers. Try each one using the two-stage approach: identify which function acts first, apply it, then apply the second.

6. Exam Technique and Common Mistakes

Most marks lost on composite function questions come from one of two errors: applying the functions in the wrong order, or making an algebraic slip when substituting an expression (rather than a number) into a function.

Composite functions bring together function notation, substitution, and algebraic manipulation — three skills that appear across the paper. Mastering the order rule for $fg(x)$ means you can handle any combination of functions the exam presents.