### About Inverse of a Function:

Every one-to-one function has an inverse. To find the inverse of a function, we start by writing f(x) as y. Then we swap the x and y variables. Now, we solve for y. Lastly, we replace y with a special notation: f-1(x), which is read as f inverse of x.

Test Objectives
• Demonstrate the ability to find the inverse of a function
Inverse of a Function Practice Test:

#1:

Instructions: find the inverse of each function.

$$a)\hspace{.2em}f(x)=2x + 1$$

$$b)\hspace{.2em}f(x)=\sqrt{x}- 2$$

#2:

Instructions: find the inverse of each function.

$$a)\hspace{.2em}f(x)=\frac{3}{7}x - \frac{6}{7}$$

$$b)\hspace{.2em}f(x)=\frac{1}{x - 2}+ 2$$

#3:

Instructions: find the inverse of each function.

$$a)\hspace{.2em}f(x)=-2 + (x - 1)^3$$

$$b)\hspace{.2em}f(x)=-x^3 - 1$$

#4:

Instructions: find the inverse of each function.

$$a)\hspace{.2em}f(x)=\sqrt{x}+ 2$$

$$b)\hspace{.2em}f(x)=\frac{2}{x - 2}$$

#5:

Instructions: find the inverse of each function.

$$a)\hspace{.2em}f(x)=\frac{x - 2}{3x - 1}$$

$$b)\hspace{.2em}f(x)=\frac{x - 1}{2x - 4}$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}f^{-1}(x)=\frac{x - 1}{2}$$

$$b)\hspace{.2em}f^{-1}(x)=(x + 2)^3$$

#2:

Solutions:

$$a)\hspace{.2em}f^{-1}(x)=\frac{7x + 6}{3}$$

$$b)\hspace{.2em}f^{-1}(x)=\frac{2x - 3}{x - 2}$$

#3:

Solutions:

$$a)\hspace{.2em}f^{-1}(x)=\sqrt{x + 2}+ 1$$

$$b)\hspace{.2em}f^{-1}(x)=\sqrt{-x - 1}$$

#4:

Solutions:

$$a)\hspace{.2em}f^{-1}(x)=(x - 2)^5$$

$$b)\hspace{.2em}f^{-1}(x)=\frac{2x + 2}{x}$$

#5:

Solutions:

$$a)\hspace{.2em}f^{-1}(x)=-\frac{x - 2}{1 - 3x}$$

$$b)\hspace{.2em}f^{-1}(x)=-\frac{4x - 1}{1 - 2x}$$