Finding the inverse of a function

How to define inverse functions

In this lesson we’ll look at the definition of an inverse function and how to find a function’s inverse.

If you remember from the last lesson, a function is invertible (has an inverse) if it’s one-to-one. Now let’s look a little more into how to find an inverse and what an inverse does.

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When you have a function with points $(x, f(x))$, the inverse function will have points $(f(x), x)$. The inverse of a function $f(x)$ is written as $f^{-1}(x)$.

For example, if $g(x)$ and $g^{-1}(x)$ are inverses of one another, then the tables below would give sets of points from each,

Now let’s look at the graphs of a function and its inverse. Look at the graph of the function $f(x)=x^3$ and its inverse $f^{-1}(x)= \sqrt[3]{x}$.

$x=(x^3)^\frac13=\sqrt[3]{x^3}$

Notice how the coordinates of the $x$ and $y$-values have switched places. Now let’s look at how to calculate an inverse algebraically.

Calculating and graphing inverse functions

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Finding the inverse of a given function

Example

What is the inverse of the function?

$f(x)=\frac{2}{3}x-4$

First, remember the function is invertible because non-horizontal linear functions are one-to-one.

To find an inverse, first rewrite $f(x)$ with the variable $y$.

$y=\frac{2}{3}x-4$

Now switch the $x$ and $y$ values.

$x=\frac{2}{3}y-4$

Now solve for $y$.

$\frac{3}{2} \cdot x+\frac{3}{2} \cdot 4=\frac{3}{2} \cdot \frac{2}{3}y$

$\frac{3}{2}x+6=y$

Now you can write the inverse function.

$f^{-1}(x)= \frac{3}{2}x+6$

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Let’s do one more example.

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When you have a function with points (x, f(x)), the inverse function will have points (f(x), x).

Example

Find the inverse of the function.

$g(x)=\frac{x}{x-3}$

First replace $g(x)$ with $y$.

$y=\frac{x}{x-3}$

Now switch the $x$ and $y$ values and solve for $y$.

$x=\frac{y}{y-3}$

$x(y-3)=y$

$xy-3x=y$

$xy-y=3x$

$y(x-1)=3x$

$y=\frac{3x}{x-1}$

The inverse function is

$g^{-1}(x) =\frac{3x}{x-1}$ 

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