If [tex]f(x)=5x[/tex], what is [tex]f^{-1}(x)[/tex]?
Answer (1)
Replace f ( x ) with y : y = 5 x .
Swap x and y : x = 5 y .
Solve for y : y = f r a c x 5 .
The inverse function is f − 1 ( x ) = 5 1 x .
Explanation
Understanding the Problem We are given the function f ( x ) = 5 x and we want to find its inverse, denoted as f − 1 ( x ) . The inverse function essentially reverses the operation of the original function.
Finding the Inverse To find the inverse function, we can follow these steps:
Replace f ( x ) with y : y = 5 x .
Swap x and y : x = 5 y .
Solve for y in terms of x .
Solving for y Now, let's solve for y in the equation x = 5 y . To isolate y , we divide both sides of the equation by 5:
5 x = 5 5 y
5 x = y
So, y = 5 x .
The Inverse Function Therefore, the inverse function is f − 1 ( x ) = 5 x , which can also be written as f − 1 ( x ) = 5 1 x .
Examples
Imagine you're converting temperatures from Celsius to Fahrenheit using a function. The inverse function would then convert Fahrenheit back to Celsius. Similarly, if you have a function that doubles the amount of money you invest, the inverse function would calculate how much you initially invested. Understanding inverse functions helps reverse processes and solve for original values in various real-world scenarios.
