Select the correct answer.

Which function is the inverse of $f(x)=x^2-16$ if the domain of $f(x)$ is $x \geq 0$?
A. $f^{-1}(x)=\sqrt{x+16}$
B. $f^{-1}(x)=\sqrt{x}+4$
C. $f^{-1}(x)=\sqrt{x-16}$
D. $f^{-1}(x)=\sqrt{x}-4

Replace f ( x ) with y : y = x 2 − 16 .
Swap x and y : x = y 2 − 16 .
Solve for y : y = ± x + 16 ​ .
Choose the positive root since the domain of f ( x ) is x ≥ 0 : f − 1 ( x ) = x + 16 ​ .
f − 1 ( x ) = x + 16 ​ ​

Explanation

Problem Analysis We are given the function f ( x ) = x 2 − 16 with the domain = "0"> x " >= "0 . Our goal is to find the inverse function f − 1 ( x ) .

Replace f(x) with y To find the inverse function, we first replace f ( x ) with y :
y = x 2 − 16

Swap x and y Next, we swap x and y :
x = y 2 − 16

Solve for y Now, we solve for y :
y 2 = x + 16 y = ± x + 16 ​

Consider the Domain Since the domain of f ( x ) is x ≥ 0 , the range of the inverse function f − 1 ( x ) must also be non-negative. Therefore, we take the positive square root: y = x + 16 ​

The Inverse Function Thus, the inverse function is: f − 1 ( x ) = x + 16 ​

Select the Correct Option Comparing this with the given options, we see that option A matches our derived inverse function.

Final Answer Therefore, the correct answer is A.


Examples
In cryptography, inverse functions are used for decryption. If f ( x ) is an encryption function, then f − 1 ( x ) is the decryption function. For example, if f ( x ) = x 2 − 16 (with =" 0"> x " >= "0 ) encrypts a message, then f − 1 ( x ) = x + 16 ​ decrypts it, allowing you to recover the original message. Understanding inverse functions is crucial for secure communication and data protection.

Answered by GinnyAnswer | 2025-07-07

The inverse of the function f ( x ) = x 2 − 16 with the domain x ≥ 0 is f − 1 ( x ) = x + 16 ​ , which corresponds to option A. This is because, when deriving the inverse, we take into account the non-negative range due to the domain restriction. Therefore, the answer is A.
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Answered by Anonymous | 2025-07-16