A function f takes an input and follows some steps to make an output. The inverse function, written f^{-1}, undoes those steps — it reverses the process, turning the output back into the original input.

Think of f as "put on your socks, then your shoes." The inverse runs it backwards, in reverse order: "take off your shoes, then your socks." Whatever f did, f^{-1} takes you straight back. It is the opposite idea to composing functions, where you chain steps forward.

Undoing the steps

Suppose f(x) = 2x + 1: it doubles the input, then adds 1. To undo it, reverse each step in reverse order — subtract 1, then halve:

Check it: f(3) = 2(3) + 1 = 7, and f^{-1}(7) = \frac{7 - 1}{2} = 3. We are right back where we started.

A recipe for the inverse

Reversing the steps in your head works for simple functions, but there is a reliable method that always works:

1. Write y = f(x).
2. Swap x and y — this captures the idea that input and output trade places.
3. Solve for y (this is just rearranging the formula).
4. That y is f^{-1}(x).

For f(x) = 2x + 1: start with y = 2x + 1, swap to get x = 2y + 1, then solve: y = \frac{x - 1}{2} — the same inverse as before.

The graph: a mirror in the line y = x

Swapping x and y has a beautiful picture. Every point (a, b) on f becomes the point (b, a) on f^{-1}. That is exactly a reflection in the line y = x: fold the page along that dashed line and the two graphs land on top of each other.

Below, f(x) = 2x + 1 and its inverse f^{-1}(x) = \frac{x - 1}{2} mirror each other across y = x.

Where f crosses the y-axis, its inverse crosses the x-axis — input and output have swapped, so the domain and range swap too.

Khan Academy introduces inverse functions here: