How to Find an Inverse Function
Inverse functions reverse each other’s work. If doubling then adding one is f(x) = 2x + 1, the inverse subtracts one then halves: f⁻¹(x) = (x−1)/2. Graphically, an inverse is the mirror image across the line y = x.
The Swap-and-Solve Method
- Write y = f(x).
- Swap every x with y (and y with x).
- Solve the new equation for y — that y is f⁻¹(x).
- Verify by composing: f(f⁻¹(x)) must equal x.
Famous Inverse Pairs
Powers pair with roots, exponentials pair with logarithms (e^x ↔ ln x), and 1/x is its own inverse. Recognizing the pairs saves algebra.
Check an inverse by composing both ways with our function composition calculator or solve the swap step with the algebra calculator.