Inverse functions
Objective. Reverse one-to-one functions and verify composition.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
An inverse function f⁻¹(x) reverses what f(x) does, so that f⁻¹(f(x)) = x for a one-to-one function. To find f⁻¹(x), write y = f(x), swap x and y, then rearrange to make y the subject again. For f(x) = 2x + 5, write y = 2x + 5, swap to x = 2y + 5, then rearrange: y = (x − 5)÷2, so f⁻¹(x) = (x − 5)÷2. You can verify an inverse by checking that f(f⁻¹(x)) simplifies back to x. Inverse functions only exist cleanly for one-to-one functions, where each output comes from exactly one input.
Worked example
Find the inverse function f⁻¹(x) of f(x) = 4x − 7.
- 1Write y = 4x − 7
- 2Swap x and y: x = 4y − 7
- 3Rearrange: x + 7 = 4y, so y = (x + 7)÷4
- 4Therefore f⁻¹(x) = (x + 7)÷4
Answer: f⁻¹(x) = (x + 7)/4
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Start practisingCommon mistake. Pupils find the inverse by simply flipping the function upside down, for example turning f(x)=3x+2 into f⁻¹(x)=1/(3x+2) rather than swapping variables and rearranging. Always use the swap-x-and-y method and check with composition.
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