Functions: composite and inverse
A function is a rule that turns an input into an output. f(x) = 3x − 2 means: take x, times it by 3, take away 2. Composite functions chain two rules together, and an inverse function runs a rule backwards.
Composite and inverse functions are on the Higher tier papers for AQA, Edexcel and OCR.
Part 1 of 3: Learn it
In short
- f(4) means put 4 in place of x.
- fg(x) means do g first, then f: fg(x) = f(g(x)).
- To find f⁻¹(x), write y = f(x), make x the subject, then swap y for x.
Where this is in your specification
Spec points: DfE A7 (composite and inverse functions are Higher tier)
| Board | Topic: Functions and graph transformations |
|---|---|
| DfE content | A7, A12, A13, A14, G21 |
| AQA 8300 | Algebra |
| Edexcel 1MA1 | Algebra |
| OCR J560 | 7 Graphs of equations and functions |
| Eduqas C300 | HA7, HA12 to HA16 |
| Cambridge IGCSE 0580 | E2.10, E2.11, E2.13 |
| National 5 C847 75 | Algebraic skills: functional notation, quadratic graphs |
Function notation
If f(x) = x² − 5, then f(3) = 9 − 5 = 4 and f(−2) = 4 − 5 = −1. To solve f(x) = 20, set x² − 5 = 20 and solve: x = 5 or x = −5.
Composite functions
Read fg(x) from right to left: the function nearest x acts first. With f(x) = 3x − 2 and g(x) = x² + 1:
- fg(x) = f(x² + 1) = 3(x² + 1) − 2 = 3x² + 1.
- gf(x) = g(3x − 2) = (3x − 2)² + 1.
- So fg(x) and gf(x) are usually different.
Inverse functions
- Write y = f(x).
- Rearrange to make x the subject.
- Replace y with x and write it as f⁻¹(x).
The inverse undoes the function, so f⁻¹(f(x)) = x. You can use that to check: put a number in f, then the answer into f⁻¹, and you should get the number back.
f(x) = 5x + 1. Find f(−2).
Show the answer
5 × (−2) + 1 = −9.
Part 2 of 3: See it worked
Worked examples
Example 1
f(x) = 3x − 2. Find f⁻¹(x).
- y = 3x − 2
- y + 2 = 3x, so x = (y + 2)/3
- Swap: f⁻¹(x) = (x + 2)/3
Answer: f⁻¹(x) = (x + 2)/3.
Example 2
h(x) = (2x + 5)/4. Find h⁻¹(x) and check it with x = 1.
- y = (2x + 5)/4, so 4y = 2x + 5
- 2x = 4y − 5, so x = (4y − 5)/2
- h⁻¹(x) = (4x − 5)/2
- Check: h(1) = 7/4, and h⁻¹(7/4) = (7 − 5)/2 = 1
Answer: h⁻¹(x) = (4x − 5)/2.
Common mistakes
- Doing f first in fg(x). The function written next to x goes first.
- Multiplying the two functions together instead of putting one inside the other.
- Thinking f⁻¹(x) means 1/f(x).
- Forgetting brackets when substituting an expression: f(x + 1) for f(x) = 3x − 2 is 3(x + 1) − 2.
f(x) = x − 4 and g(x) = 2x. Find gf(3).
Show the answer
f(3) = −1, then g(−1) = −2.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
f(x) = 5x + 1. Find f(−2).
5 × (−2) + 1 = −9.
f(x) = x − 4 and g(x) = 2x. Find gf(3).
f(3) = −1, then g(−1) = −2.
Find the inverse of f(x) = x/3 + 6.
y − 6 = x/3, so f⁻¹(x) = 3(x − 6).
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
Full lessons and marked practice for this course are coming soon to Brainlag Learn. See courses