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 = 5 x .
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, f − 1 ( x ) . The inverse function essentially reverses the operation of the original function.
Finding the Inverse Function 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 : y = 5 x
Replace y with f − 1 ( x ) : f − 1 ( x ) = 5 x
The Inverse Function Therefore, the inverse function is f − 1 ( x ) = 5 1 x .
Examples
Imagine you're converting between miles and kilometers. If f ( x ) = 1.609 x converts miles ( x ) to kilometers, then f − 1 ( x ) = 1.609 x converts kilometers back to miles. Understanding inverse functions helps you reverse processes, like converting units, encoding/decoding messages, or undoing mathematical operations. This concept is crucial in many fields, including engineering, computer science, and cryptography.
