Determine the values of x for which f(x) and g(x) are inverse functions.
-$$f ( x ) = - x ^ { 2 } + 16 , g ( x ) = \sqrt { 16 - x }$$
A) $x \leq 16$
B) $x \geq 16$
C) $x \leq 0$
D) $x \geq 0$

Question

Quizplus · Accepted Answer

For two functions to be inverses of each other, applying one after the other in any order should return the original input. Starting with $f(x) = -x^2 + 16$, if $g(x)$ is its inverse, then $g(f(x)) = x$ and $f(g(x)) = x$. Substituting $f(x)$ into $g(x)$, we get $g(-x^2 + 16) = \sqrt{16 - (-x^2 + 16)} = \sqrt{x^2} = |x|$. Since $g(x)$ involves a square root, its domain must be $x \geq 0$ to ensure real outputs, matching the absolute value result which is always non-negative.

Determine the Values of X for Which F(x) and G(x) f(x)=−x2+16,g(x)=16−xf ( x ) = - x ^ { 2 } + 16 , g ( x ) = \sqrt { 16 - x }f(x)=−x2+16,g(x)=16−x​

Determine the Values of X for Which F(x) and G(x) $f ( x ) = - x ^ { 2 } + 16 , g ( x ) = \sqrt { 16 - x }$