An inverse function is a function that will reverse the effect produced by the original function. These functions have the main characteristic that they are a reflection of the original function with respect to the line y = x. The coordinates of the inverse function are the same as the original function, but the values of x and y are swapped.
We will look at an overview of inverse functions along with the process used to find the inverse of a function. In addition, we will see several examples with answers to understand the application of this process.
Summary of inverse functions
Inverse functions are functions that reverse the effect of the original function. The inverse of a function has the same points as the original function except that the values of x and y are swapped.
For example, if the original function contains the points (1, 2) and (-3, -5), the inverse function will contain the points (2, 1) and (-5, -3).
The inverse of $latex f(x)$ is denoted as $latex {{f}^{-1}}(x)$. Note that in the notation for inverses, the “-1” is not an exponent despite looking like one. Thus, we have to remember that:
$latex {{f}^{-1}}(x)\neq \frac{1}{f(x)}$
Finding the inverse of a function
Given the function $latex f(x)$, we can find the inverse function $latex {{f}^{- 1}}(x)$ by following these steps:
Step 1: First, substitute $latex f(x)$ with y. This helps us to facilitate the rest of the process.
Step 2: Substitute each x with a y and each y with an x.
Step 3: Solve the equation obtained in step 2 for y.
Step 4: Replace y by $latex {{f}^{-1}}$ since this is the inverse function.
Inverse functions – Examples with answers
The process to find inverse functions is applied to solve the following examples. Each example has its step-by-step solution to learn how to find inverse functions. It is recommended to try to solve the exercises first before looking at the solution.
EXAMPLE 1
Find the inverse of the function $latex f(x)=2x-3$.
Solution
Step 1: Substitute $latex f(x)$ with y:
$latex f(x)=2x-3$
$latex y=2x-3$
Step 2: Swap the variables x and y:
$latex x=2y-3$
Step 3: Solve the equation for y:
$latex x=2y-3$
$latex x+3=2y$
$latex \frac{x+3}{2}=y$
Step 4: Substitute y with $latex {{f}^{-1}}(x)$:
$latex {{f}^{-1}}(x)=\frac{x+3}{2}$
EXAMPLE 2
If we have the function $latex f(x)=3x+4$, find $latex {{f}^{- 1}}(x)$.
Solution
Step 1: We start substituting $latex f(x)$ with y to facilitate the process:
$latex f(x)=3x+4$
$latex y=3x+4$
Step 2: Substitute the x with the y and the y with the x:
$latex x=3y+4$
Step 3: Solve for y in the equation from step 2:
$latex x=3y+4$
$latex x-4=3y$
$latex \frac{x-4}{3}=y$
Step 4: We swap y with $latex {{f}^{-1}}(x)$:
$latex {{f}^{-1}}(x)=\frac{x-4}{3}$
EXAMPLE 3
What is the inverse of the function $latex f(x)=\frac{x+ 4}{2x-5}$?
Solution
Step 1: We have to substitute $latex f(x)$ with y to facilitate the process:
$latex f(x)=\frac{x+4}{2x-5}$
$latex y=\frac{x+4}{2x-5}$
Step 2: We swap the variables x and y:
$latex x=\frac{y+4}{2y-5}$
Step 3: We solve for y in the equation from step 2:
$latex x=\frac{y+4}{2y-5}$
$latex x(2y-5)=y+4$
$latex 2xy-5x=y+4$
$latex 2xy-y=5x+4$
$latex y(2x-1)=5x+4$
$latex y=\frac{5x+4}{2x-1}$
Step 4: We have to substitute $latex {{f}^{-1}}(x)$ with y:
$latex {{f}^{-1}}(x)=\frac{5x+4}{2x-1}$
EXAMPLE 4
Given the function $latex f(x)=\log_{10}(x)$, find the inverse function $latex {{f}^{-1}}(x)$.
Solution
Step 1: We start substituting $latex f(x)$ with y to visualize more easily:
$latex f(x)=\log_{10}(x)$
$latex y=\log_{10}(x)$
Step 2: We substitute x with y and y with x:
$latex x=\log_{10}(y)$
Step 3: We solve the equation for y:
$latex x=\log_{10}(y)$
$latex {{10}^x}=y$
Step 4: We substitute y with $latex {{f}^{-1}}(x)$:
$latex {{f}^{-1}}(x)={{10}^x}$
EXAMPLE 5
If we have $latex f(x)=\sqrt{x-5}$, find $latex {{f}^{-1}}(x)$.
Solution
Step 1: We start swapping $latex f(x)$ with y:
$latex f(x)=\sqrt{x-5}$
$latex y=\sqrt{x-5}$
Step 2: We swap the variables x and y:
$latex x=\sqrt{y-5}$
Step 3: We solve for y by squaring both sides of the equation:
$latex x=\sqrt{y-5}$
$latex {{x}^2}=y-5$
$latex {{x}^2}+5=y$
Step 4: We substitute y with $latex {{f}^{-1}}(x)$:
$latex {{f}^{-1}}(x)={{x}^2}+5$
EXAMPLE 6
Find the inverse function of $latex f(x)=\frac{x+1}{x}$.
Solution
Step 1: We substitute $latex f(x)$ with y:
$latex f(x)=\frac{x+1}{x}$
$latex y=\frac{x+1}{x}$
Step 2: We substitute x with y and y with x:
$latex x=\frac{y+1}{y}$
Step 3: We solve for y:
$latex x=\frac{y+1}{y}$
$latex xy=y+1$
$latex xy-y=1$
$latex y(x-1)=1$
$latex y=\frac{1}{x-1}$
Step 4: We subsitute y with $latex {{f}^{-1}}(x)$:
$latex {{f}^{-1}}(x)=\frac{1}{x-1}$
EXAMPLE 7
Given the function $latex f(x)=\sqrt[4]{2x+5}$, find its inverse function.
Solution
Step 1: We substitute $latex f(x)$ with y:
$latex f(x)=\sqrt[4]{2x+5}$
$latex y=\sqrt[4]{2x+5}$
Step 2: We swap the variables x and y:
$latex x=\sqrt[4]{2y+5}$
Step 3: We raise both sides to the power of 4 in order to solve for y:
$latex x=\sqrt[4]{2y+5}$
$latex {{x}^4}=2y+5$
$latex {{x}^4}-5=2y$
$latex \frac{{{x}^4}-5}{2}=y$
Step 4: We substitute y with $latex {{f}^{-1}}(x)$:
$latex {{f}^{-1}}(x)=\frac{{{x}^4}-5}{2}$
Inverse functions – Practice problems
Put your knowledge of inverse functions into practice with the following problems. If you need help with this, you can look at the process used in the examples above.
See also
Interested in learning more about functions? Take a look at these pages:
Jefferson Huera Guzman
Jefferson is the lead author and administrator of Neurochispas.com. The interactive Mathematics and Physics content that I have created has helped many students.
