Determine whether the functions are inverses of each other.
\[
\begin{array}{ll}
f(x)=x^{2}+7, & \text { domain }[0, \infty) \\
g(x)=\sqrt{x-7}, & \text { domain }[7, \infty)
\end{array}
\]
Are $f$ and $g$ inverses of each other?
No
Yes

Step 1 ：We are given two functions, \(f(x) = x^{2} + 7\) with domain \([0, \infty)\) and \(g(x) = \sqrt{x - 7}\) with domain \([7, \infty)\). We are asked to determine whether these functions are inverses of each other.

Step 2 ：Two functions are inverses of each other if and only if the composition of the two functions in both orders results in the identity function. That is, if \(f\) and \(g\) are inverses, then \(f(g(x)) = x\) for all \(x\) in the domain of \(g\), and \(g(f(x)) = x\) for all \(x\) in the domain of \(f\).

Step 3 ：We can check whether \(f(g(x)) = x\) and \(g(f(x)) = x\) for the given functions \(f\) and \(g\).

Step 4 ：Let's first compute \(f(g(x))\). Substituting \(g(x) = \sqrt{x - 7}\) into \(f(x)\), we get \(f(g(x)) = (\sqrt{x - 7})^{2} + 7 = x\).

Step 5 ：Next, let's compute \(g(f(x))\). Substituting \(f(x) = x^{2} + 7\) into \(g(x)\), we get \(g(f(x)) = \sqrt{(x^{2} + 7) - 7} = \sqrt{x^{2}} = x\).

Step 6 ：Since both \(f(g(x))\) and \(g(f(x))\) simplify to \(x\), we conclude that \(f\) and \(g\) are indeed inverses of each other.

Step 7 ：\(\boxed{\text{Yes, } f \text{ and } g \text{ are inverses of each other.}}\)