What's the inverse of [tex]f(x)=x^2-16[/tex]?
A) [tex]f^{-1}(x)= \pm \sqrt{x+16}[/tex]
B) [tex]f^{-1}(x)= \pm \sqrt{x^2+16}[/tex]
C) [tex]f^{-1}(x)=x-4[/tex]
D) [tex]f(x)=x^2-16[/tex] and [tex]g(x)=x-4[/tex]
Answer (2)
Replace f ( x ) with y : y = x 2 − 16 .
Swap x and y : x = y 2 − 16 .
Solve for y : y = ± x + 16 .
The inverse function is f − 1 ( x ) = ± x + 16 .
Explanation
Understanding the Problem We are given the function f ( x ) = x 2 − 16 and we want to find its inverse f − 1 ( x ) . The inverse function is found by swapping x and y and solving for y .
Swapping x and y Let y = f ( x ) , so we have y = x 2 − 16 . To find the inverse, we swap x and y to get x = y 2 − 16 .
Isolating y^2 Now we solve for y . Add 16 to both sides of the equation: x + 16 = y 2
Solving for y Take the square root of both sides: y = ± x + 16
Finding the Inverse Function Thus, the inverse function is f − 1 ( x ) = ± x + 16 . Comparing this to the given options, we see that option A matches our result.
Examples
Understanding inverse functions is crucial in many areas, such as cryptography and data encryption. For example, if f ( x ) represents an encryption function, then f − 1 ( x ) would be the decryption function. Imagine you have a secret code where f ( x ) = x 2 − 16 encrypts a message. To decode it, you need the inverse function f − 1 ( x ) = ± x + 16 . This concept ensures secure communication by allowing only those with the inverse function to decipher the original message.
The inverse of the function f ( x ) = x 2 − 16 is f − 1 ( x ) = ± x + 16 , which corresponds to Option A. This is found by swapping x and y, isolating y, and taking the square root. Thus, the chosen answer is A.
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