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# Intro to invertible functions

Not all functions have inverses. Those who do are called "invertible." Learn how we can tell whether a function is invertible or not.

**Inverse functions**, in the most general sense, are functions that "reverse" each other. For example, if

## Do all functions have an inverse function?

Consider the finite function $h$ defined by the following table.

We can create a mapping diagram for function $h$ .

Now let's reverse the mapping to find the inverse, ${h}^{-1}$ .

Notice here that ${h}^{-1}$ maps the input of $2$ to two different outputs: $1$ and $3$ . This means that ${h}^{-1}$ is

*a function.***not**Because the inverse of $h$ is not a function, we say that $h$ is

**non-invertible**.In general, a function is

**invertible**only if each input has a unique output. That is, each output is paired with exactly one input. That way, when the mapping is reversed, it will still be a function!Here's an example of an invertible function $g$ . Notice that the inverse is indeed a function.

## Check your understanding

### Challenge Problem

## Invertible functions and their graphs

Consider the graph of the function $y={x}^{2}$ .

We know that a function is invertible if each input has a unique output. Or in other words, if each output is paired with exactly one input.

But this is not the case for $y={x}^{2}$ .

Take the output $4$ , for example. Notice that by drawing the line $y=4$ , you can see that there are two inputs, $2$ and $-2$ , associated with the output of $4$ .

In fact, if you slide the horizontal line up and down, you will see that most outputs are associated with two inputs! So the function $y={x}^{2}$ is a

**non-invertible**function.In contrast, consider the function $y={x}^{3}$ .

If we take a horizontal line and slide it up and down the graph, it only ever intersects the function in one spot!

This means that each output corresponds with exactly one input. In other words, each input has a unique output. The function $y={x}^{3}$ is invertible.

The reasoning above describes what is called the horizontal line test: In general, a function $f$ is invertible if it

*.***passes the horizontal line test**## Check your understanding

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