Which of the following functions is the correct inverse for the function [tex]$f(x)=x^2+2$[/tex]?
A. [tex]$f^{-1}(x)= \pm \sqrt{x-2}$[/tex]
B. [tex]$f^{-1}(x)=x^2-2$[/tex]
C. [tex]$f^{-1}(x)= \pm \sqrt{x}-2$[/tex]
D. [tex]$f^{-1}(x)= \pm \sqrt{x+2}$[/tex]
Answer (2)
Set y = f ( x ) = x 2 + 2 .
Solve for x in terms of y : x 2 = y − 2 , so x = ± y − 2 .
The inverse function is f − 1 ( x ) = ± x − 2 .
The correct inverse function is f − 1 ( x ) = ± x − 2 .
Explanation
Finding the Inverse Function To find the inverse of the function f ( x ) = x 2 + 2 , we need to solve for x in terms of y , where y = f ( x ) .
Isolating x^2 Let y = x 2 + 2 . We want to isolate x . Subtracting 2 from both sides gives us y − 2 = x 2 .
Solving for x Taking the square root of both sides, we get x = ± y − 2 .
The Inverse Function Therefore, the inverse function is f − 1 ( x ) = ± x − 2 .
Examples
Understanding inverse functions is crucial in many areas, such as cryptography and data encryption. For example, if f ( x ) encrypts a message, then f − 1 ( x ) decrypts it, returning the original message. In real life, this ensures secure communication over the internet, protecting sensitive information from unauthorized access. The ability to find and apply inverse functions is fundamental to maintaining privacy and security in the digital age.
The correct inverse function for f ( x ) = x 2 + 2 is f − 1 ( x ) = ± x − 2 , which is option A. This shows how for each value of x greater than or equal to 2, there can be two corresponding values of f − 1 ( x ) . The steps to find the inverse involve isolating x and solving the equation.
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