# How to find Inverse of a Function Since an inverse function essentially undoes the effects of the original function, it is important for you to learn how to use them. Therefore, in this article, we have tried to acquaint you with the method of using inverse functions.

## Definition of Function Inverses

• An inverse function is a function that reverses another function: if the function f applied to an input $$x$$ gives a result of $$y$$, then applying its inverse function $$g$$ to $$y$$ gives the result $$x$$.
$$f(x)=y$$ if and only if $$g(y)=x$$
• The inverse function of $$f(x)$$ is usually shown by $$f^{-1} (x)$$.

## Examples

### Function Inverses – Example 1:

Find the inverse of the function: $$f(x)=2x-1$$

Solution:

First, replace $$f(x)$$ with $$y: y=2x-1$$, Then, replace all $$x^{‘}s$$ with $$y$$ and all $$y^{‘}s$$ with $$x: x=2y-1$$, Now, solve for $$y: x=2y-1→x+1=2y→\frac{1}{2} x+\frac{1}{2}=y$$ Finally replace y with $$f^{-1} (x): f^{-1} (x)=\frac{1}{2} x+\frac{1}{2}$$

### Function Inverses – Example 2:

Find the inverse of the function: $$g(x)=\frac{1}{5} x+3$$

Solution:

$$g(x)=\frac{1}{5} x+3→y=\frac{1}{5} x+3→$$ replace all $$x^{‘}s$$ with y and all $$y^{‘}s$$ with $$x$$
$$x=\frac{1}{5} y+3$$, solve for $$y: →x-3=\frac{1}{5} y→5(x-3)=y→g^{-1}(x)=5x-15$$

### Function Inverses – Example 3:

Find the inverse of the function: $$h(x)=\sqrt{x}+6$$

Solution:

$$h(x)=\sqrt{x}+6→y=\sqrt{x}+6$$, replace all $$x^{‘}s$$ with y and all $$y^{‘}s$$ with $$x →x=\sqrt{y}+6→x-6=\sqrt{y}→(x-6)^2=\sqrt{y}^2→x^2-12x+36=y →h^{-1} (x)=x^2-12x+36$$

### Function Inverses – Example 4:

Find the inverse of the function: $$g(x)=\frac{x+5}{4}$$

Solution:

$$g(x)=\frac{x+5}{4}→y=\frac{x+5}{4}→$$ replace all $$x^{‘}s$$ with y and all $$y^{‘}s$$ with $$x$$
$$x=\frac{y+5}{4}$$, solve for $$y: →4x=y+5→4x-5=y→g^{-1}(x)=4x-5$$

## Exercises for Function Inverses

### Find the inverse of each function.

1. $$\color{blue}{f(x)=\frac{1}{x}-3}$$
$$\color{blue}{f^{-1} (x)=}$$________
2. $$\color{blue}{g(x)=2x^3-5}$$
$$\color{blue}{g^{-1} (x)=}$$________
3. $$\color{blue}{h(x)=10x}$$
$$\color{blue}{h^{-1} (x)=}$$________
4. $$\color{blue}{f(x)=\sqrt{x}-4}$$
$$\color{blue}{f^{-1} (x)=}$$________
5. $$\color{blue}{f(x)=3x^2+2}$$
$$\color{blue}{f^{-1} (x)=}$$________
6. $$\color{blue}{h(x)=22x}$$
$$\color{blue}{h^{-1} (x)=}$$________
1. $$\color{blue}{\frac{1}{x+3}}$$
2. $$\color{blue}{\sqrt{\frac{x+5}{2}}}$$
3. $$\color{blue}{\frac{x}{10}}$$
4. $$\color{blue}{x^2+8x+16}$$
5. $$\color{blue}{\sqrt{\frac{x-2}{3}}}$$
6. $$\color{blue}{\frac{x}{22}}$$ 36% OFF

X

## How Does It Work? ### 1. Find eBooks

Locate the eBook you wish to purchase by searching for the test or title.  ### 3. Checkout

Complete the quick and easy checkout process. ## Why Buy eBook From Effortlessmath? Save up to 70% compared to print  Help save the environment  Over 2,000 Test Prep titles available Over 80,000 happy customers Over 10,000 reviews with an average rating of 4.5 out of 5  