Inverse Functions (2)

f^-1 undoes f. Swap x and y, solve for y.

The idea

The inverse of a function $f$ is the function that recovers each input from its output.

Definition (Inverse function).

The inverse of $f$, written $f^{-1}$, is the function defined by the two round trips $f^{-1}(f(x)) = x \quad \text{and} \quad f(f^{-1}(x)) = x.$

The $-1$ is part of the name, not an exponent: $f^{-1}$ is not $1/f$.

To find an inverse, write $y = f(x)$ and solve for $x$ in terms of $y$; the formula says how to get $y$ from $x$, and the solved equation says how to get $x$ back from $y$. Then rename the letters so the input is again called $x$ — or, equivalently, swap $x$ and $y$ at the start and solve for $y$.

Not every function has an inverse. If two inputs share an output, the reverse rule would have to send that output back to both, and a function may not.

Theorem.

A function $f$ has an inverse if and only if it is one-to-one: distinct inputs give distinct outputs.

Because inverting swaps input with output, the point $(a, b)$ on the graph of $f$ becomes $(b, a)$ on the graph of $f^{-1}$, and the two graphs are mirror images across the line $y = x$.

Ways to work on it

• Walkthrough. What an inverse function is, and the swap-and-solve method for finding one.
• Practice. Invert a random linear function.
• Hardest. Invert non-linear functions such as powers and reciprocals.

Not sure where to start? Take the ten-question placement test.