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## Algebra (all content)

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

Lesson 20: Finding inverse functions (Algebra 2 level)

# Finding inverse functions

Learn how to find the formula of the inverse function of a given function. For example, find the inverse of f(x)=3x+2.
Inverse functions, in the most general sense, are functions that "reverse" each other. For example, if $f$ takes $a$ to $b$, then the inverse, ${f}^{-1}$, must take $b$ to $a$.
Or in other words, $f\left(a\right)=b\phantom{\rule{0.278em}{0ex}}⟺\phantom{\rule{0.278em}{0ex}}{f}^{-1}\left(b\right)=a$.
In this article we will learn how to find the formula of the inverse function when we have the formula of the original function.

## Before we start...

In this lesson, we will find the inverse function of $f\left(x\right)=3x+2$.
Before we do that, let's first think about how we would find ${f}^{-1}\left(8\right)$.
To find ${f}^{-1}\left(8\right)$, we need to find the input of $f$ that corresponds to an output of $8$. This is because if ${f}^{-1}\left(8\right)=x$, then by definition of inverses, $f\left(x\right)=8$.
$\begin{array}{rl}f\left(x\right)& =3x+2\\ \\ 8& =3x+2& & \text{Let f(x)=8}\\ \\ 6& =3x& & \text{Subtract 2 from both sides}\\ \\ 2& =x& & \text{Divide both sides by 3}\end{array}$
So $f\left(2\right)=8$ which means that ${f}^{-1}\left(8\right)=2$

## Finding inverse functions

We can generalize what we did above to find ${f}^{-1}\left(y\right)$ for any $y$.
To find ${f}^{-1}\left(y\right)$, we can find the input of $f$ that corresponds to an output of $y$. This is because if ${f}^{-1}\left(y\right)=x$ then by definition of inverses, $f\left(x\right)=y$.
$\begin{array}{rl}f\left(x\right)& =3x+2\\ \\ y& =3x+2& & \text{Let f(x)=y}\\ \\ y-2& =3x& & \text{Subtract 2 from both sides}\\ \\ \frac{y-2}{3}& =x& & \text{Divide both sides by 3}\end{array}$
So ${f}^{-1}\left(y\right)=\frac{y-2}{3}$.
Since the choice of the variable is arbitrary, we can write this as ${f}^{-1}\left(x\right)=\frac{x-2}{3}$.

### 1) Linear function

Find the inverse of $g\left(x\right)=2x-5$.
${g}^{-1}\left(x\right)=$

### 2) Cubic function

Find the inverse of $h\left(x\right)={x}^{3}+2$.
${h}^{-1}\left(x\right)=$

### 3) Cube-root function

Find the inverse of $f\left(x\right)=4\cdot \sqrt{x}$.
${f}^{-1}\left(x\right)=$

### 4) Rational functions

Find the inverse of $g\left(x\right)=\frac{x-3}{x-2}$.
${g}^{-1}\left(x\right)=$

### 5) Challenge problem

Match each function with the type of its inverse.