If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### 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 $f\left(x\right)=4\cdot \sqrt[3]{\phantom{A}x}$.
${f}^{-1}\left(x\right)=$

### 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.

## Want to join the conversation?

• Why are rational functions called "rational"?
• Rational numbers are numbers that can be expressed as a fraction (a ratio) of two integers.
Rational functions are also fractions (ratios) - with polynomials in the numerator and denominator.
• What is the inverse of the function e to the power of x? Or some other exponential function?
• How do you find the inverse of y=x+3/3
?
• y=(x+3)/3
3y=x+3
3y-3=x
∴x=3y-3
• how do you find the inverse of F(x)= x/(x-1)
• 𝑦 = 𝑥/(𝑥 – 1)
Swap the domain and range:
𝑥 = 𝑦/(𝑦 – 1)
Now solve for the inverse:
𝑥𝑦 – 𝑥 = 𝑦
𝑥𝑦 – 𝑦 = 𝑥
𝑦(𝑥 – 1) = 𝑥
𝑦 = 𝑥/(𝑥 – 1)
So interestingly, this function is its own inverse. Functions like this are called involutions. Comment if you have questions!
• x-1 = -1/y+1, I got stuck on this part, I don't know how to get rid of a fraction with a denominator like that. help me please!
• I assume y+1 is the denominator, not just y?

We take your equation and multiply both sides by y+1. On the right, we get -1(y+1)/(y+1), and the (y+1)'s cancel. So we're left with
(x-1)(y+1)= -1
Now we divide by (x+1), then subtract 1 to isolate y.
• How do you find the inverse with a set of numbers?