Functions appear across the higher tier papers, and composite and inverse functions are among the trickiest — but also most rewarding — topics once you understand the logic. All three major boards test this: AQA (8300) typically places it in Paper 2 or 3, Edexcel (1MA1) on any of the three papers, and OCR (J560) usually in Section B of the higher tier. Get this right and you are looking at 3–6 marks that many students leave on the table.
What You Need to Know First
- A function takes an input, applies a rule, and produces an output. We write f(x) to mean "the function f applied to x."
- You need to be confident substituting values into expressions — if f(x) = 3x + 1, then f(4) = 3(4) + 1 = 13.
- You must be able to rearrange equations to change the subject — this is essential for finding inverse functions.
- Know the difference between f(x), f(2), and f(g(x)) before you continue.
- A negative or fractional input is perfectly valid — do not panic when you see f(−3).
Composite Functions: f(g(x))
A composite function means applying one function and then feeding the result straight into another. The notation fg(x) or f(g(x)) means "do g first, then apply f to the answer."
The golden rule: work from the inside out. Whatever is in the innermost bracket, evaluate that first.
Worked Example 1
Let f(x) = 2x + 3 and g(x) = x² − 1.
Find fg(4).
Step 1: Start with g(4).
g(4) = 4² − 1 = 16 − 1 = 15
Step 2: Now apply f to that result.
f(15) = 2(15) + 3 = 30 + 3 = 33
So fg(4) = 33.
Find gf(4).
Step 1: Start with f(4).
f(4) = 2(4) + 3 = 8 + 3 = 11
Step 2: Apply g to 11.
g(11) = 11² − 1 = 121 − 1 = 120
So gf(4) = 120.
Notice that fg(4) ≠ gf(4). Order matters — composite functions are not commutative.
Writing fg(x) as a Single Expression
Sometimes the question asks you to find fg(x) in terms of x rather than for a specific value.
Let f(x) = 5x − 2 and g(x) = x + 4.
Find fg(x).
Step 1: Replace every x in f(x) with the expression g(x).
fg(x) = f(x + 4) = 5(x + 4) − 2
Step 2: Expand and simplify.
= 5x + 20 − 2
= 5x + 18
To find gf(x):
gf(x) = g(5x − 2) = (5x − 2) + 4 = 5x + 2
Again, different answers — always check which order the question is asking for.
Inverse Functions: Reversing the Machine
The inverse function f⁻¹(x) undoes what f(x) does. If f takes an input x and gives an output y, then f⁻¹ takes y and gives back x.
Think of it like this: f(x) is a machine that converts pounds to dollars. f⁻¹(x) is the same machine run in reverse, converting dollars back to pounds.
How to Find an Inverse Function
Step 1: Write f(x) = y (replace f(x) with y).
Step 2: Swap x and y (write x where y was, and y where x was).
Step 3: Rearrange to make y the subject.
Step 4: Replace y with f⁻¹(x).
Worked Example 2
Let f(x) = 4x − 7. Find f⁻¹(x).
Step 1: Write as y = 4x − 7.
Step 2: Swap x and y: x = 4y − 7.
Step 3: Rearrange for y:
x + 7 = 4y
y = (x + 7) / 4
Step 4: Write the answer:
f⁻¹(x) = (x + 7) / 4
Check your answer: f(3) = 4(3) − 7 = 5. Then f⁻¹(5) = (5 + 7) / 4 = 12 / 4 = 3. ✓ It gives back the original input.
Worked Example 3 — Harder Case
Let g(x) = (2x + 1) / (x − 3). Find g⁻¹(x).
Step 1: y = (2x + 1) / (x − 3).
Step 2: Swap: x = (2y + 1) / (y − 3).
Step 3: Multiply both sides by (y − 3):
x(y − 3) = 2y + 1
xy − 3x = 2y + 1
Collect all y terms on one side:
xy − 2y = 3x + 1
y(x − 2) = 3x + 1
Divide both sides by (x − 2):
y = (3x + 1) / (x − 2)
Step 4: g⁻¹(x) = (3x + 1) / (x − 2)
This type appears at the top end of higher tier papers and can be worth 3–4 marks on its own.
Common Mistakes Students Make
- Doing composite functions in the wrong order. fg(x) means do g first — students often apply f first. Always read right to left for the order of operations.
- Forgetting to substitute the whole expression. If f(x) = x² and g(x) = x + 1, then fg(x) = f(x + 1) = (x + 1)², not x² + 1. You must square the whole bracket.
- Confusing f⁻¹(x) with 1/f(x). The ⁻¹ notation here means the inverse function, not a reciprocal. f⁻¹(x) ≠ 1/f(x).
- Forgetting to swap x and y. This is the step students most often skip. If you do not swap, you end up rearranging the original function and get f(x) back.
- Not checking the answer. You can always verify an inverse by checking that f(f⁻¹(a)) = a for a simple value. One extra line of working that earns you confidence.
Exam Tips for Functions
- Write down every step. Even if you get the final answer wrong, method marks (M marks) are available for showing the correct process — especially on the inverse function method.
- Label which function you are working with. Write "g(4) = ..." then "f(15) = ..." clearly. Examiners award marks line by line.
- Check the question says fg or gf — the order is everything. Mis-reading this is one of the most common single-mark losses on this topic.
- On Edexcel, leaving an answer unsimplified often loses the accuracy (A) mark. Always expand and simplify composite function expressions fully.
- AQA awards the method mark for swapping x and y correctly, even if the subsequent rearrangement has an arithmetic error. So always show the swap step explicitly.
Practise Now
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