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.

What Is a Composite Function?

A composite function applies one function to the result of another. The notation fg(x) means "apply g first, then apply f to the result". This is sometimes written as f(g(x)).

The order matters: fg(x) and gf(x) usually give different results.

Key Formulas

Think of it as working from the inside out: the function closest to x is applied first.

Step-by-Step Method

1. Identify which function is applied first (the inner function, closest to x).
2. Write down the expression for the inner function.
3. Substitute that entire expression into the outer function, replacing every x.
4. Simplify the result.
5. To evaluate at a number (e.g., fg(2)), either substitute 2 into your composite expression or find g(2) first, then apply f to the result.

Worked Example 1 — Foundation Level

Question: f(x) = 3x + 1 and g(x) = x². Find fg(2).

Working:

Step 1: fg(2) means apply g first, then f.

Step 2: Find g(2) = 2² = 4.

Step 3: Apply f to the result: f(4) = 3(4) + 1 = 13.

Answer: fg(2) = 13

Worked Example 2 — Higher Level

Question: f(x) = 2x − 5 and g(x) = x² + 1. Find expressions for (a) fg(x) and (b) gf(x).

Working:

(a) fg(x) = f(g(x)) = f(x² + 1)

Replace x in f with (x² + 1): 2(x² + 1) − 5 = 2x² + 2 − 5 = 2x² − 3.

(b) gf(x) = g(f(x)) = g(2x − 5)

Replace x in g with (2x − 5): (2x − 5)² + 1 = 4x² − 20x + 25 + 1 = 4x² − 20x + 26.

Answer: (a) fg(x) = 2x² − 3, (b) gf(x) = 4x² − 20x + 26

Worked Example 3 — Exam Style

Question: f(x) = 4x − 3 and g(x) = x/2 + 1. (a) Find ff(x). (b) Solve fg(x) = 17.

Working:

(a) ff(x) = f(f(x)) = f(4x − 3)

Replace x in f with (4x − 3): 4(4x − 3) − 3 = 16x − 12 − 3 = 16x − 15.

(b) First find fg(x):

fg(x) = f(g(x)) = f(x/2 + 1) = 4(x/2 + 1) − 3 = 2x + 4 − 3 = 2x + 1.

Set 2x + 1 = 17: 2x = 16, so x = 8.

Check: g(8) = 4 + 1 = 5. f(5) = 20 − 3 = 17 ✓

Answer: (a) ff(x) = 16x − 15, (b) x = 8

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.

Practice Questions

Q1 (Higher): f(x) = 5x − 2 and g(x) = x + 4. Find fg(3).

Q2 (Higher): f(x) = x² and g(x) = 2x + 3. Find an expression for gf(x).

Q3 (Higher): f(x) = 3x + 1 and g(x) = x − 2. Solve gf(x) = 10.

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Related Topics

• Functions and Function Notation
• Inverse and Composite Functions
• Solving Linear Equations

Summary

• A composite function applies one function to the result of another: fg(x) = f(g(x)).
• Always apply the inner function (closest to x) first, then the outer function.
• fg(x) and gf(x) are generally different — the order matters.
• ff(x) means applying the same function twice: f(f(x)).
• To solve fg(x) = k, find the composite expression first, then solve the equation.