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Additional Mathematics 06061.1

Composite and inverse functions

Combining functions and reversing them.

Learning objectives

What you need to be able to do

Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.

  • 1.1.1Understand domain and range, and find composite and inverse functions.

6 minute read

Composite and inverse functions

A function maps each input to exactly one output. The domain is the set of allowed inputs; the range is the set of outputs produced.

Composite functions

fg(x) means "do g first, then f". The function written closer to x acts first. If f(x) = 2x + 1 and g(x) = x²:

  • fg(x) = f(x²) = 2x² + 1
  • gf(x) = g(2x + 1) = (2x + 1)²

These are different, so order matters — a very common source of lost marks.

Inverse functions

f⁻¹(x) undoes f. To find it:

  1. Write y = f(x).
  2. Swap x and y.
  3. Rearrange to make y the subject.
  4. That expression is f⁻¹(x).

For f(x) = 3x − 4: y = 3x − 4 → x = 3y − 4 → y = (x + 4)/3, so f⁻¹(x) = (x + 4)/3.

Two facts worth knowing

  • ff⁻¹(x) = x — applying a function then its inverse returns the original. This is a quick way to check your inverse.
  • The graph of f⁻¹ is the reflection of f in the line y = x.
  • Only one-to-one functions have inverses. f(x) = x² has no inverse over all real numbers, because 4 maps back to both 2 and −2 — which is why the domain is often restricted to x ≥ 0.

Domain and range in practice

Watch for values that break the function: division by zero, and square roots of negatives. For f(x) = 1/(x − 2), the domain excludes x = 2.

Think of it like this

A composite function is a production line: the item passes through the inner machine first. An inverse is running the line backwards, which only works if no two inputs ever produced the same output.

Worked examples

Method, step by step

Given f(x) = 3x − 4 and g(x) = x + 2, find fg(x) and f⁻¹(x).

  1. 1fg(x) means apply g first: g(x) = x + 2.
  2. 2Then apply f to that: f(x + 2) = 3(x + 2) − 4 = 3x + 6 − 4 = 3x + 2.
  3. 3For the inverse, write y = 3x − 4 and swap: x = 3y − 4.
  4. 4Rearrange: 3y = x + 4, so y = (x + 4)/3.

fg(x) = 3x + 2 and f⁻¹(x) = (x + 4)/3

Common misconceptions

  • Reading `fg(x)` as "f first". The inner function g acts first.
  • Thinking `f⁻¹(x)` means `1/f(x)`. The −1 denotes the inverse function, not a reciprocal.
  • Assuming every function has an inverse. Only one-to-one functions do, which is why domains get restricted.

In the exam

  • Check an inverse by computing `ff⁻¹(x)`; if it does not simplify to x, the inverse is wrong.
  • When asked for a domain, look specifically for division by zero and square roots of negative numbers.