How to Find Inverse of a Function? (+FREE Worksheet!)

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:

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

Function Inverses – Example 3:

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

Solution:

First, replace \(h(x)\) with \(y:\) \(y=\sqrt{x}+6\), then, replace all \(x^{‘}s\) with y and all \(y^{‘}s\) with \(x : x=\sqrt{y}+6\), now, solve for \(y :\) \(x=\sqrt{y}+6\) → \(x-6=\sqrt{y}→(x-6)^2=\sqrt{y}^2→x^2-12x+36=y\) , Finally replace \(y\) with \(h^{-1}(x): h^{-1} (x)=x^2-12x+36\)

Function Inverses – Example 4:

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

Solution:

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

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