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

How to Find the Inverse of a Function


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 an example.
Example 1: 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. $$x - 2=(y - 1)^3$$ $$\sqrt[3]{x - 2}=y - 1$$ $$y=\sqrt[3]{x - 2}+ 1$$ Step 4) Replace y with f-1(x) $$f^{-1}(x)=\sqrt[3]{x - 2}+ 1$$

Skills Check:

Example #1

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

Please choose the best answer.

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. $$g(x)=-x^5$$

Please choose the best answer.

A
$$g^{-1}(x)=-\sqrt[5]{x}$$
B
$$g^{-1}(x)=\sqrt[5]{x}- 2$$
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. $$f(x)=-\frac{1}{x - 1}+ 3$$

Please choose the best answer.

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
$$f^{-1}(x)=\frac{3}{x}$$
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