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### Course: Algebra (all content) > Unit 7

Lesson 19: Introduction to inverses of functions (Algebra 2 level)# Intro to inverse functions

Learn what the inverse of a function is, and how to evaluate inverses of functions that are given in tables or graphs.

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

For example, here we see that function $f$ takes $1$ to $x$ , $2$ to $z$ , and $3$ to $y$ .

The inverse of $f$ , denoted ${f}^{-1}$ (and read as "$f$ inverse"), will ${f}^{-1}$ takes $x$ to $1$ , $y$ to $3$ , and $z$ to $2$ .

*reverse*this mapping. Function## Defining inverse functions

In general, if a function $f$ takes $a$ to $b$ , then the inverse function, ${f}^{-1}$ , takes $b$ to $a$ .

From this, we have the formal definition of inverse functions:

$f(a)=b{\textstyle \phantom{\rule{0.278em}{0ex}}}\u27fa{\textstyle \phantom{\rule{0.278em}{0ex}}}{f}^{-1}(b)=a$

Let's dig further into this definition by working through a couple of examples.

### Example 1: Mapping diagram

Suppose function $h$ is defined by mapping diagram above. What is ${h}^{-1}(9)$ ?

### Solution

We are given information about function $h$ and are asked a question about function ${h}^{-1}$ . Since inverse functions reverse each other, we need to

*reverse*our thinking.Specifically, to find ${h}^{-1}(9)$ , we can find the input of $h$ whose output is $9$ . This is because if ${h}^{-1}(9)=x$ , then by definition of inverses, $h(x)=9$ .

From the mapping diagram, we see that $h(6)=9$ , and so ${h}^{-1}(9)=6$ .

### Check your understanding

### Example 2: Graph

This is the graph of function $g$ . Let's find ${g}^{-1}(-7)$ .

### Solution

To find ${g}^{-1}(-7)$ , we can find the input of $g$ that corresponds to an output of $-7$ . This is because if ${g}^{-1}(-7)=x$ , then by definition of inverses, $g(x)=-7$ .

From the graph, we see that $g(-3)=-7$ .

Therefore, ${g}^{-1}(-7)=-3$ .

### Check your understanding

## A graphical connection

The examples above have shown us the algebraic connection between a function and its inverse, but there is also a graphical connection!

Consider function $f$ , given in the graph and in a table of values.

We can reverse the inputs and outputs of function $f$ to find the inputs and outputs of function ${f}^{-1}$ . So if $(a,b)$ is on the graph of $y=f(x)$ , then $(b,a)$ will be on the graph of $y={f}^{-1}(x)$ .

This gives us these graph and table of values of ${f}^{-1}$ .

Looking at the graphs together, we see that the graph of $y=f(x)$ and the graph of $y={f}^{-1}(x)$ are reflections across the line $y=x$ .

This will be true in general; the graph of a function and its inverse are reflections over the line $y=x$ .

## Check your understanding

## Why study inverses?

It may seem arbitrary to be interested in inverse functions but in fact we use them all the time!

Consider that the equation $C={\displaystyle \frac{5}{9}}(F-32)$ can be used to convert the temperature in degrees Fahrenheit, $F$ , to a temperature in degrees Celsius, $C$ .

But suppose we wanted an equation that did the reverse – that converted a temperature in degrees Celsius to a temperature in degrees Fahrenheit. This describes the function $F={\displaystyle \frac{9}{5}}C+32$ , or the inverse function.

On a more basic level, we solve many equations in mathematics, by "isolating the variable". When we isolate the variable, we "undo" what is around it. In this way, we are using the idea of inverse functions to solve equations.

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