1. Determine if the two functions are inverses.
\[
\begin{array}{l}
g(x)=\sqrt{9 x-8}+16 \\
\text { and } \\
g^{-1}(x)=\frac{(-16+x)^{2}+8}{9} \text { where } x \geq 16
\end{array}
\]
Yes, these are inverse functions.
No, these are not inverse functions.
Answer
\(\boxed{\text{Yes, these are inverse functions}}\)
Step 1 ：Given the two functions \(g(x)=\sqrt{9x-8}+16\) and \(g^{-1}(x)=\frac{(-16+x)^{2}+8}{9}\) where \(x \geq 16\)
Step 2 ：To determine if these two functions are inverses of each other, we substitute \(g^{-1}(x)\) into \(g(x)\) and see if we get \(x\)
Step 3 ：Substituting \(g^{-1}(x)\) into \(g(x)\), we get \(\sqrt{(x - 16)^{2}} + 16\)
Step 4 ：The result of this substitution is \(x\), which is the identity function
Step 5 ：Since the result is the identity function, the two functions are inverses of each other
Step 6 ：\(\boxed{\text{Yes, these are inverse functions}}\)
