Main content

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

Lesson 20: Finding inverse functions (Algebra 2 level)- Finding inverse functions: linear
- Finding inverse functions: quadratic
- Finding inverse functions: quadratic (example 2)
- Finding inverse functions: radical
- Finding inverses of rational functions
- Finding inverse functions
- Finding inverses of linear functions

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# 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

Or in other words, $f(a)=b{\textstyle \phantom{\rule{0.278em}{0ex}}}\u27fa{\textstyle \phantom{\rule{0.278em}{0ex}}}{f}^{-1}(b)=a$ .

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 $f(x)=3x+2$ .

Before we do that, let's first think about how we would find ${f}^{-1}(8)$ .

To find ${f}^{-1}(8)$ , we need to find the input of $f$ that corresponds to an output of $8$ . This is because if ${f}^{-1}(8)=x$ , then by definition of inverses, $f(x)=8$ .

So $f(2)=8$ which means that ${f}^{-1}(8)=2$

## Finding inverse functions

We can generalize what we did above to find ${f}^{-1}(y)$ for any $y$ .

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

So ${f}^{-1}(y)={\displaystyle \frac{y-2}{3}}$ .

Since the choice of the variable is arbitrary, we can write this as ${f}^{-1}(x)={\displaystyle \frac{x-2}{3}}$ .

## Check your understanding

### 1) Linear function

### 2) Cubic function

### 3) Cube-root function

### 4) Rational functions

### 5) Challenge problem

## Want to join the conversation?

- Why are rational functions called "rational"?(17 votes)
- 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.(28 votes)

- What is the inverse of the function e to the power of x? Or some other exponential function?

Thank you in advance.(10 votes) - How do you find the inverse of y=x+3/3

?(5 votes)- y=(x+3)/3

3y=x+3

3y-3=x

∴x=3y-3(5 votes)

- how do you find the inverse of F(x)= x/(x-1)(2 votes)
- 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!(10 votes)

- 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!(3 votes)
- 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.(2 votes)

- How would you find the inverse of a function where x is greater/less than a specific number? Example problem: Write the inverse of f(x) = 2x+3 where x > 5(2 votes)
- What is the inverse function of f(x)= x*2 - x(3 votes)
- On question number 4, I don't understand why, we subtract x on from the right, instead of subtracting 'xy' from the left. We need to solve for 'x' right?

Can't solve it both ways?(3 votes) - this question is related to FUNCTIONS topic from maths

How to solve f^1(1/2)

when the only information given in the question is

f(x)=3/2x-5(2 votes) - How do you find the inverse with a set of numbers?(2 votes)