Lesson Objectives
• Learn how to find the inverse of a one-to-one function

## How to Find the Inverse of a Function

Before we discuss the concept of an inverse function, let's quickly review what we know about functions and one-to-one functions. A function is a rule that assigns a unique output value (y-value) to each input value (x-value). In other words, for each x-value, there is only one corresponding y-value. For a function to have an inverse, it must be a one-to-one function. In other words, each input value (x-value) in the domain must correspond to a unique output value (y-value) in the range, and vice versa. This property ensures that there are no repetitions or ambiguities in the mapping. In other words, for a function to be a one-to-one function, each x-value corresponds to exactly one y-value and each y-value corresponds to exactly one x-value.
The inverse of a function f, denoted as f-1, essentially reverses the mapping of the original one-to-one function. It swaps the roles of the input (x-value) and output (y-value) values. Instead of mapping from the domain to the range, the inverse function maps from the range back to the domain. By applying the inverse function to the output value of the original one-to-one function, we can retrieve the corresponding input value. If what you just read was a bit confusing, don't worry we will discuss it in more detail in the next lesson. For now, just think about a function and its inverse as undoing each other. In other words, the goal of finding an inverse is to find a function that, when composed with the original function, returns the input value unchanged. This composition is expressed as f(f-1(x)) = x, which demonstrates the idea that the function and its inverse cancel out each other when applied together.
To think about how to find the inverse of a one-to-one function, let's start with a simple example.
Example #1: Find the inverse of F. $$F=\{(3, 2), (4, 5), (7, 8), (-1, -2)\}$$ To find the inverse of our one-to-one function F, we will simply swap the x and y values. In other words, the domain becomes the range and the range becomes the domain. Let's represent our inverse function with G. $$G=\{(2, 3), (5, 4), (8, 7), (-2, -1)\}$$ As the previous simple example shows, the inverse of a one-to-one function is found by simply interchanging the x and y values. When we think about finding the inverse of a function defined by y = f(x), we can use a similar procedure.

### Finding the Equation of the Inverse of y = f(x)

For a one-to-one function defined by an equation y = f(x), we can find the inverse f-1(x) using the following steps:
• Replace f(x) with y
• Interchange x and y in the equation
• Solve for y
• Replace y with f-1(x)
Let's look at a few examples.
Example 2: Find the inverse of each function. $$f(x)=(x - 1)^3 + 2$$ Step 1) Replace f(x) with y. $$y=(x - 1)^3 + 2$$ Step 2) Interchange x and y in the equation. $$x=(y - 1)^3 + 2$$ Step 3) Solve for y. $$x=(y - 1)^3 + 2$$ Subtract 2 away from each side: $$x - 2=(y - 1)^3$$ Take the cube root of each side: $$\sqrt{x - 2}=y - 1$$ Add 1 to each side: $$y=\sqrt{x - 2}+ 1$$ Step 4) Replace y with f-1(x) $$f^{-1}(x)=\sqrt{x - 2}+ 1$$ Example 3: Find the inverse of each function. $$f(x)=\frac{2}{x - 3}+ 1$$ Step 1) Replace f(x) with y. $$y=\frac{2}{x - 3}+ 1$$ Step 2) Interchange x and y in the equation. $$x=\frac{2}{y - 3}+ 1$$ Step 3) Solve for y. $$x=\frac{2}{y - 3}+ 1$$ Subtract 1 away from each side: $$x - 1=\frac{2}{y - 3}$$ Multiply both sides by (y - 3)/(x - 1): $$\require{cancel}\frac{(y - 3)}{\cancel{(x - 1)}}\cdot \cancel{(x - 1)}=\frac{2}{\cancel{(y - 3)}}\cdot \frac{\cancel{(y - 3)}}{(x - 1)}$$ $$y - 3=\frac{2}{x - 1}$$ Add 3 to each side: $$y=\frac{2}{x - 1}+ 3$$ Get a common denominator (optional): $$y=\frac{2}{x - 1}+ \frac{3(x - 1)}{x - 1}$$ $$y=\frac{2}{x - 1}+ \frac{3x - 3}{x - 1}$$ $$y=\frac{2 + 3x - 3}{x - 1}$$ $$y=\frac{3x - 1}{x - 1}$$ Step 4) Replace y with f-1(x) $$f^{-1}(x)=\frac{3x - 1}{x - 1}$$

#### Skills Check:

Example #1

Find the inverse of each. $$g(x)=x - 6$$

A
$$g^{-1}(x)=\frac{4x + 3}{3}$$
B
$$g^{-1}(x)=\frac{1}{2}x$$
C
$$g^{-1}(x)=x + 6$$
D
$$g^{-1}(x)=\frac{1}{2}x - \frac{5}{2}$$
E
$$g^{-1}(x)=\frac{x}{6}$$

Example #2

Find the inverse of each. $$g(x)=-x^5$$

A
$$g^{-1}(x)=-\sqrt{x}$$
B
$$g^{-1}(x)=\sqrt{x}- 2$$
C
$$g^{-1}(x)=-2x^2 + 2$$
D
$$g^{-1}(x)=-\frac{x}{5}$$
E
$$g^{-1}(x)=-5x^2 - 2$$

Example #3

Find the inverse of each. $$f(x)=-\frac{1}{x - 1}+ 3$$

A
$$f^{-1}(x)=\frac{3}{x}- 2$$
B
$$f^{-1}(x)=-\frac{1}{x - 3}+ 1$$
C
$$f^{-1}(x)=-\frac{3}{x + 2}- 2$$
D
$$f^{-1}(x)=-\frac{3}{-x-3}$$
E
$$f^{-1}(x)=\frac{3}{x}$$         