Inverse Functions

An inverse function is simply the "undo" button for another function: whatever $f$ does, $f^{-1}$ takes you back. Converting Celsius to Fahrenheit has an inverse, converting Fahrenheit back to Celsius. The formula up top says exactly that and nothing more: $f^{-1}(f(x)) = x$ (do $f$, then undo it, and you're back to $x$), and the same the other way round. The $-1$ is just a label meaning "inverse," not an exponent. You'll use inverses whenever you know the answer and want to work back to the input.

One-to-one. Undoing only works if you can tell where you came from. If $f(2) = 4$ and $f(-2) = 4$, then given $4$, the inverse wouldn't know whether to send it back to $2$ or $-2$. So a function needs to be one-to-one, meaning different inputs always give different outputs. On a graph, that means no horizontal line hits it twice. $x^2$ fails, so it has no inverse unless you restrict its domain, say to $x \ge 0$.

Finding an inverse. An inverse swaps the roles of input and output, so that's literally what you do:

1. Write $y = f(x)$.
2. Swap $x$ and $y$.
3. Solve for $y$; that's $f^{-1}(x)$.
4. Check that $f(f^{-1}(x)) = x$.

The same swap explains the graph: every point $(a, b)$ on $f$ becomes $(b, a)$ on $f^{-1}$, which is a reflection across the line $y = x$. And the domain of $f^{-1}$ is the range of $f$, and vice versa, since inputs and outputs have traded places.

A common mistake. After the exponent rules, it's tempting to read $f^{-1}(x)$ as $\frac{1}{f(x)}$. It isn't. For $f(x) = 3x - 5$, the inverse is $\frac{x + 5}{3}$, while $\frac{1}{f(x)} = \frac{1}{3x - 5}$ is a completely different function.

Decrypting undoes encrypting; converting Celsius back to Fahrenheit undoes the forward conversion; logarithms undo exponentials to solve for time in growth problems.