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### 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

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

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