**f**relates an input

**x**to an output

**f(x)**...

...an inverse function

**f**relates the output

^{−1}**f(x)**back to the input

**x**:

Imagine a function

**f**relates an input

**2**to an output

**3**...

f(2) = 3

...the inverse function **f**relates

^{−1}**3**back to

**2**...

f

An inverse function is denoted ^{−1}(3) = 2**f**.

^{−1}(x)## A Real Example of an Inverse Function

It is easier to understand inverse functions with an example. Let us consider a function:## f(x) = x + 1 and Its Inverse

The function**f(x) = x + 1**takes each input and adds 1 to it. The mapping diagram below shows this function.

Now let's consider the inverse function of

**f(x) = x + 1**. The inverse function will take us back to the original values. The inverse function takes each input and subtracts 1 from it. The inverse of

**f(x) = x + 1**is:

## The Inverse Function Takes Us Back to the Input

We can combine a function with its inverse.- Imagine we put a value into a function and then put the result into the inverse function. The output is original value.

If we put an input**x**into a function**f(x)**, and then put the result into the inverse function**f**, the output is^{−1}(x)**x**.

- Imagine we put a value into the inverse function and then put the result into the function. The output is original value.

If we put an input**x**into a function**f(x)**, and then put the result into the inverse function**f**, the output is^{−1}(x)**x**.

## Graphs of Inverse Functions

Functions and their inverses can be plotted on graphs.- Functions relate an input
**x**to an output**y**. We can plot these pairs of inputs and outputs as Cartesian coordinates,**(x, y)**. - The inverse function relates the output
**y**back to the input**x**. We swap the coordinates around to find the inverse function, such that**(x, y)**becomes**(y, x)**.

**y = x**.

## The Horizontal Line Test

For the inverse function to be a function, each input can only be related to one output. This means that for the function (which will be reflected in**y = x**), each value of

**y**can only be related to one value of

**x**. On a graph, this means that any horizontal line only crosses the curve once. This is the

*horizontal line test*. The curve below passes the horizontal line test. Any horizontal line that is drawn will cross the curve only once. The inverse will be a function:The curve below does not pass the horizontal line test. A horizontal line can be drawn which crosses the curve twice. The inverse will not be a function:

## You might also like...

functionsevaluating a composite functionfinding the inverse of a functionfinding the inverse of a function using a graph

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