Composite and Inverse Functions — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers function notation and the two ways of building new functions from old ones: composing them, by feeding the output of one into another, and inverting them, by running a function backwards. By the end of this guide you should be able to evaluate a function at a number or an expression, form and use composite functions, and find the inverse of a linear function.

You should also be able to read the order of a composite correctly, solve equations written in function notation, and recognise that a function and its inverse undo one another.

The organising idea is that a function is a machine: something goes in, a rule is applied, something comes out. Composing two functions means wiring two machines in series, so the order matters — and the notation records that order in a way that trips people up, because fg(x) means the g machine runs first. Inverting a function means running the machine backwards, undoing each operation in the reverse order, exactly as when solving an equation. Both halves of the topic are that one picture.

Key terms and definitions

Function — a rule giving exactly one output for each input, written f(x).

Input and output — the value put in, and the value that comes out.

Evaluate — substitute a value into the function and work out the result.

Composite function — a function made by applying one function to the output of another, written fg(x) or gf(x).

Inverse function — the function that reverses f, written f⁻¹(x).

Self-inverse — a function that is its own inverse, such as f(x) = 1/x.

Core concepts

Function notation

f(x) = 2x + 3 defines a rule: double the input, then add 3.

To evaluate, replace every x with the given value. So f(5) = 2(5) + 3 = 13, and f(0) = 3.

The input does not have to be a number. f(a) = 2a + 3, and f(x + 1) = 2(x + 1) + 3 = 2x + 5. Substituting an expression requires brackets, and forgetting them is a common source of error.

Note that f(5) does not mean f multiplied by 5. The notation looks like multiplication but is not.

Solving equations in function notation

A question such as "solve f(x) = 17 for f(x) = 2x + 3" is asking which input produces that output.

Set the rule equal to the value and solve: 2x + 3 = 17, so 2x = 14 and x = 7.

This is the same skill as solving any linear equation, dressed in new notation.

Composite functions and the order trap

A composite applies one function to the output of another.

fg(x) means do g first, then f. The function written closest to the x is the one that acts first, which is the opposite of how the letters read from left to right. This is the single most important fact in the topic, and it is where most marks are lost.

With f(x) = 2x and g(x) = x + 1:

fg(x) = f(g(x)) = f(x + 1) = 2(x + 1) = 2x + 2.

gf(x) = g(f(x)) = g(2x) = 2x + 1.

These are different functions, so fg(x) and gf(x) are generally not equal. Checking one against the other is a quick way to confirm you have read the order correctly.

Composites with numbers

There are two equally valid routes, and the step-by-step one is safer.

To find fg(3) with the functions above: first g(3) = 3 + 1 = 4, then f(4) = 2 × 4 = 8.

Alternatively, substitute 3 into the composite formula fg(x) = 2x + 2, giving 8 as well.

Working in two steps avoids the algebra entirely and makes the order visible, so use it unless the question asks for the composite as an expression.

Inverse functions

The inverse reverses the function: it takes an output and returns the input that produced it.

The standard method has three steps. Write y = f(x). Rearrange to make x the subject. Then swap the letters, writing the result as f⁻¹(x).

For f(x) = 2x + 1: write y = 2x + 1, subtract 1 to get y − 1 = 2x, divide to get x = (y − 1)/2, then write f⁻¹(x) = (x − 1)/2.

The rearranging is exactly the equation-solving skill from earlier in the course: undo the operations in the reverse order they were applied. Here the rule doubled and then added 1, so the inverse subtracts 1 and then halves.

Note that f⁻¹ does not mean 1 over f. The superscript −1 marks an inverse function, not a reciprocal.

Checking an inverse

Applying a function and then its inverse returns you to where you started, so ff⁻¹(x) = x.

For the example above, f⁻¹(7) = (7 − 1)/2 = 3, and f(3) = 2(3) + 1 = 7. ✓

This check takes seconds and catches almost every rearranging error.

Finding an inverse at a single value

Sometimes only one value is wanted, and the full inverse is unnecessary.

To find f⁻¹(8) when f(x) = 4x, ask what input gives an output of 8. Since 4x = 8, the answer is 2.

Reading f⁻¹(8) as "the input that produced 8" turns the question into a one-line equation.

Worked examples

Example 1: A composite, both ways round

For f(x) = 3x and g(x) = x + 4, find fg(2) and gf(2).

For fg(2), g acts first: g(2) = 2 + 4 = 6. Then f(6) = 3 × 6 = 18.

For gf(2), f acts first: f(2) = 3 × 2 = 6. Then g(6) = 6 + 4 = 10.

The two answers differ, which is the normal situation. If they had come out the same it would be worth re-checking the order.

Example 2: Finding an inverse

Find the inverse of f(x) = 5x − 2.

Write y = 5x − 2.

Rearrange for x: add 2 to give y + 2 = 5x, then divide by 5 to give x = (y + 2)/5.

Write the answer in terms of x: f⁻¹(x) = (x + 2)/5.

Check: f(3) = 15 − 2 = 13, and f⁻¹(13) = 15/5 = 3. ✓ The original rule multiplied then subtracted, and the inverse adds then divides — the same operations undone in reverse order.

Example 3: A composite as an expression

For f(x) = x² and g(x) = x + 3, find gf(x) and fg(x).

For gf(x), f acts first, giving x². Then g adds 3: gf(x) = x² + 3.

For fg(x), g acts first, giving x + 3. Then f squares the whole thing: fg(x) = (x + 3)².

The brackets are essential. Writing x² + 3 for fg(x) would square only the x, which is a different function entirely — at x = 1 the two give 16 and 4.

Common mistakes and how to avoid them

Reading fg(x) left to right. The function nearer the x goes first, so fg means g then f.

Assuming fg(x) equals gf(x). They are usually different functions.

Treating f⁻¹ as a reciprocal. It means the inverse function, not 1 over f.

Forgetting brackets when substituting an expression. For f(x) = x², f(x + 3) is (x + 3)², not x² + 3.

Leaving the inverse in terms of y. Finish by swapping back so the answer is written as f⁻¹(x).

Undoing the operations in the wrong order. The inverse reverses the sequence, so the last operation applied is the first one undone.

Reading f(5) as multiplication. It means the function applied to 5.

Exam technique for "Composite and Inverse Functions"

Write the order out before calculating a composite. A single note — "g first, then f" — prevents the error that costs most marks here.

Work composites with numbers in two separate steps and show both. Each step can earn credit, and the intermediate value makes the order visible to the examiner.

For an inverse, show all three stages: y = f(x), the rearrangement, and the final answer in f⁻¹(x) form. The rearrangement is where the method marks are.

Check every inverse by substituting a number through both functions. It takes one line and confirms the answer completely.

Keep brackets around any expression substituted into a function, especially where a squaring or a multiplication follows.

When only one value of an inverse is wanted, set the original function equal to that value and solve — there is no need to find the whole inverse.

Quick revision summary

A function is a machine: f(x) = 2x + 3 doubles the input and adds 3, so f(5) = 13. Substituting an expression needs brackets.

fg(x) means do g first, then f — the function nearest the x acts first. So with f(x) = 2x and g(x) = x + 1, fg(x) = 2x + 2 while gf(x) = 2x + 1. The two are generally different.

With numbers, work a composite in two steps: find the inner value, then apply the outer function.

To find an inverse: write y = f(x), rearrange to make x the subject, then rewrite it as f⁻¹(x). The inverse undoes the operations in the reverse order, so f(x) = 2x + 1 gives f⁻¹(x) = (x − 1)/2.

f⁻¹ means the inverse function, never a reciprocal.

Check an inverse by putting a number through the function and then through the inverse: you should get back to where you started.

To find f⁻¹ at a single value, ask which input gives that output and solve the resulting equation.