Inverse Functions

A function turns inputs into outputs. Often you need to go the other way: from a Celsius reading back to Fahrenheit, from a price with tax back to the price before tax. The function that runs a process backward is its inverse function. This page shows what an inverse is, when one exists, how to justify that with specific values, and how to find it from a table, a graph or a formula.

What an inverse does

If $f(a) = b$, the inverse function $f^{-1}$ sends $b$ back to $a$: $f^{-1}(b) = a$. Input and output trade places. Read $f^{-1}(x)$ as "f inverse of x".

The $-1$ is part of the name. It does not mean a reciprocal: $f^{-1}(x)$ and $\frac{1}{f(x)}$ are different things. If $f(x) = x - 5$, then $f^{-1}(x) = x + 5$, while $\frac{1}{f(x)} = \frac{1}{x - 5}$.

Undoing and redoing gets you back where you started, so composing a function with its inverse gives the identity function: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. This is the best way to check an inverse.

When does an inverse exist?

An inverse must send each output back to one input. If two inputs share an output, say $h(1) = 1$ and $h(3) = 1$, then $h^{-1}(1)$ would have to be both $1$ and $3$. A rule that does that is not a function, so $h$ has no inverse.

A function in which every output comes from exactly one input is called one-to-one, or invertible. Functions that are always increasing, or always decreasing, are invertible.

Exam readers give no credit for a bare phrase such as "it fails the horizontal line test" or "it is not one-to-one". Name the inputs and the shared output. And say "$f$ has no inverse", not "$f$ is not an inverse".

Inverses from tables and graphs

From a table, swap the rows: the outputs of $f$ become the inputs of $f^{-1}$. If $f(3) = 5$, then $f^{-1}(5) = 3$. The inputs you can use in $f^{-1}$ are exactly the outputs of $f$.

On a graph, swapping coordinates turns each point $(a, b)$ into $(b, a)$. That is a reflection across the line $y = x$. To read $f^{-1}(b)$ from the graph of $f$, find $b$ on the output axis, go across to the graph, and read the input below it.

Inverses from formulas

Write $y = f(x)$, swap $x$ and $y$, and solve for $y$. Another way to think of it: list what $f$ does to $x$, then undo the steps in reverse order.

Restricting the domain

A function that is not invertible can become invertible on part of its domain. A parabola rises on one side of its vertex and falls on the other, so on either side alone no output repeats.

Domain, range and context

In context, say what the inverse's input and output mean. For $C(F) = \frac{5(F - 32)}{9}$, $C^{-1}(20) = 68$ means that $20$ degrees Celsius is $68$ degrees Fahrenheit.