Every function is a journey. You feed in an input, the function marches it through its steps, and out comes an output. The inverse function, written f^{-1}, is the journey home: it takes the output and walks it back, step by step, to exactly the input you started with.

You already use inverse pairs every day. Wrap a gift — the inverse is unwrapping it. Convert a temperature from °C to °F — the inverse converts it back from °F to °C. Encrypt a message — the inverse decrypts it. Put on your socks, then your shoes — the inverse takes off your shoes, then your socks, undoing each step in reverse order. Whatever f did, f^{-1} undoes it exactly. It is the opposite idea to composing functions, where you chain steps forward: here we run the chain backward.

In fact, "undoes it exactly" has a tidy composite-function meaning: f^{-1} \circ f is the do-nothing function. Go out, come home, and you are standing where you began.

Undoing the steps

Suppose f(x) = 2x + 1. As a machine, it does two things to its input, in order:

1. double it,
2. then add 1.

To undo the journey, reverse each step, in reverse order — just like the shoes and socks:

1. subtract 1 (undoing the "add 1"),
2. then halve (undoing the "double").

Check it: f(3) = 2(3) + 1 = 7, and f^{-1}(7) = \frac{7 - 1}{2} = 3. We are right back where we started. Notice the order mattered: "halve, then subtract 1" would give \tfrac{7}{2} - 1 = 2.5 — not home at all.

The swap-and-rearrange ritual

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

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

Worked example. Find the inverse of f(x) = 3x + 2.

Now verify it really is the journey home. Send 5 out and bring it back:

So f^{-1}\bigl(f(5)\bigr) = 5. Always do this check in an exam — it takes ten seconds and catches almost every slip.

A harder one: a fraction

The ritual shines when the steps are too tangled to reverse in your head. Find the inverse of

Write y = \frac{2}{x-3}, swap, and solve:

Verify: f(4) = \frac{2}{4-3} = 2, and f^{-1}(2) = \frac{2}{2} + 3 = 4. Home again.

Notice how the forbidden values swapped along with everything else: f can't accept x = 3 and never outputs 2 \cdot \frac{1}{x-3} = 0… so f^{-1} can't accept 0 and never outputs 3. The domain and range of a function and its inverse always trade places — inputs and outputs have literally swapped jobs.

The graph: a mirror in the line y = x

Swapping x and y has a beautiful picture. If f(1) = 3, the point (1, 3) sits on the graph of f — and since f^{-1}(3) = 1, the point (3, 1) sits on the graph of f^{-1}. Every point (a, b) on f becomes the point (b, a) on f^{-1}, and swapping coordinates like 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.

Read the mirror like a detective. Where f crosses the y-axis at (0, 1), its inverse crosses the x-axis at (1, 0) — input and output have swapped. A steep stretch of f becomes a shallow stretch of f^{-1} (gradient 2 flips to gradient \tfrac{1}{2}). And the one place the two curves can meet on the mirror is where f(x) = x — here at (-1, -1), a point that the journey out doesn't move at all.

This picture is also an exam shortcut: asked to sketch an inverse, you don't need its formula — just reflect the graph you have in y = x.

When there's no way home: x²

Try the ritual on g(x) = x^2 and something breaks. From x = y^2 we get y = \pm\sqrt{x} — that \pm means two answers for one input, and a function is only allowed one. The picture shows why:

The horizontal line y = 4 cuts the parabola twice: both 2 and -2 square to 4. So the "journey home from 4" has two possible destinations, and no function can choose both. g is many-to-one, and many-to-one functions have no inverse. (This is the horizontal line test: if any horizontal line crosses the graph more than once, no inverse exists.)

The fix is surgery: cut the domain in half. Keep only x \geq 0 — the right-hand arm of the parabola — and now every output comes from exactly one input. The restricted function is one-to-one, and its inverse is an old friend:

The half-parabola and the square-root curve are perfect mirror images in y = x. This is exactly why your calculator's \sqrt{\phantom{x}} button only ever returns the positive root — it is the inverse of the restricted square, so it must pick one destination. The same trick powers the inverse trig functions you'll meet later: \sin^{-1} only exists because someone cut sine's domain down to a one-to-one slice.

Prefer to hear it in another voice? Khan Academy's introduction covers the same journey-home idea: