To determine whether [tex]\( F(x) \)[/tex] and [tex]\( G(x) \)[/tex] are inverse functions across the domain [tex]\([5, +\infty)\)[/tex], we need to check if the composition [tex]\( F(G(x)) \)[/tex] results in the original input [tex]\( x \)[/tex] for [tex]\( x \)[/tex] in this domain.
Given the functions:
[tex]\[ F(x) = \sqrt{x-5} + 4 \][/tex]
[tex]\[ G(x) = (x-4)^2 + 5 \][/tex]
We need to compute [tex]\( F(G(x)) \)[/tex] and see if it simplifies to [tex]\( x \)[/tex].
First, compute [tex]\( G(x) \)[/tex]:
[tex]\[ G(x) = (x-4)^2 + 5 \][/tex]
Now, substitute [tex]\( G(x) \)[/tex] into [tex]\( F(x) \)[/tex]:
[tex]\[ F(G(x)) = F((x-4)^2 + 5) \][/tex]
Substitute [tex]\((x-4)^2 + 5\)[/tex] into [tex]\( F(x) \)[/tex]:
[tex]\[ F((x-4)^2 + 5) = \sqrt{((x-4)^2 + 5) - 5} + 4 \][/tex]
[tex]\[ = \sqrt{(x-4)^2} + 4 \][/tex]
Since [tex]\(\sqrt{(x-4)^2} = |x-4|\)[/tex], this simplifies to:
[tex]\[ F((x-4)^2 + 5) = |x-4| + 4 \][/tex]
For [tex]\( x \in [5, +\infty) \)[/tex], [tex]\( x-4 \geq 1 \)[/tex] and thus [tex]\(|x-4| = x-4\)[/tex]. Therefore:
[tex]\[ F((x-4)^2 + 5) = x - 4 + 4 \][/tex]
Simplifying, we get:
[tex]\[ F(G(x)) = x \][/tex]
So, the composition [tex]\( F(G(x)) = x \)[/tex], implying that [tex]\( F(x) \)[/tex] and [tex]\( G(x) \)[/tex] exhibit an inverse relationship within the domain [tex]\([5, +\infty)\)[/tex].
From the given answer choices, the correct one is:
[tex]\[
\boxed{D. No, because \sqrt{G(x)-5}+4 \neq x}
\][/tex]
