If [tex]$f(x)=x-2$[/tex], which of the following is the inverse of [tex]$f(x)$[/tex]?
A. [tex]$f^{-1}(x)=2-x$[/tex]
B. [tex]$f^{-1}(x)=2 x$[/tex]
C. [tex]$f^{-1}(x)=x-2$[/tex]
D. [tex]$f^{-1}(x)=x+2$[/tex]
Answer (1)
Replace f ( x ) with y : y = x − 2 .
Swap x and y : x = y − 2 .
Solve for y : y = x + 2 .
The inverse function is f − 1 ( x ) = x + 2 .
Explanation
Understanding the Problem We are given the function f ( x ) = x − 2 and asked 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, we can follow these steps:
Replace f ( x ) with y : y = x − 2 .
Swap x and y : x = y − 2 .
Solve for y : y = x + 2 .
Replace y with f − 1 ( x ) : f − 1 ( x ) = x + 2 .
Identifying the Correct Option Therefore, the inverse of f ( x ) = x − 2 is f − 1 ( x ) = x + 2 . Comparing this to the given options, we see that option D matches our result.
Examples
In real life, inverse functions can be used to convert between different units of measurement. For example, if f ( x ) converts Celsius to Fahrenheit, then f − 1 ( x ) converts Fahrenheit back to Celsius. Understanding inverse functions helps in reversing processes or calculations, which is useful in many practical situations.
