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.

## Related Topics

- How to Add and Subtract Functions
- How to Multiply and Dividing Functions
- How to Solve Function Notation
- How to Solve Composition of 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.

- \(\color{blue}{f(x)=\frac{1}{x}-3}\)

\(\color{blue}{f^{-1} (x)=}\)________ - \(\color{blue}{g(x)=2x^3-5}\)

\(\color{blue}{g^{-1} (x)=}\)________ - \(\color{blue}{h(x)=10x}\)

\(\color{blue}{h^{-1} (x)=}\)________ - \(\color{blue}{f(x)=\sqrt{x}-4}\)

\(\color{blue}{f^{-1} (x)=}\)________ - \(\color{blue}{f(x)=3x^2+2}\)

\(\color{blue}{f^{-1} (x)=}\)________ - \(\color{blue}{h(x)=22x}\)

\(\color{blue}{h^{-1} (x)=}\)________

- \(\color{blue}{\frac{1}{x+3}}\)
- \(\color{blue}{\sqrt[3]{\frac{x+5}{2}}}\)
- \(\color{blue}{\frac{x}{10}}\)
- \(\color{blue}{x^2+8x+16}\)
- \(\color{blue}{\sqrt{\frac{x-2}{3}}}\)
- \(\color{blue}{\frac{x}{22}}\)