Find $f(g(x))$ and $g(f(x))$ and determine whether the pair of functions $f$ and $g$ are inverses of each other.
\[
f(x)=\frac{9}{x-4} \text { and } g(x)=\frac{9}{x}+4
\]
a. $f(g(x))=$
b. $g(f(x))=$
c. $f$ and $g$ are inverses of each other.
$f$ and $g$ are not inverses of each other.

Step 1 ：Given the functions \(f(x)=\frac{9}{x-4}\) and \(g(x)=\frac{9}{x}+4\)

Step 2 ：Substitute \(g(x)\) into \(f(x)\) to find \(f(g(x))\)

Step 3 ：\(f(g(x))=f\left(\frac{9}{x}+4\right)=\frac{9}{\left(\frac{9}{x}+4\right)-4}\)

Step 4 ：Simplify to get \(f(g(x))=x\)

Step 5 ：Substitute \(f(x)\) into \(g(x)\) to find \(g(f(x))\)

Step 6 ：\(g(f(x))=g\left(\frac{9}{x-4}\right)=\frac{9}{\left(\frac{9}{x-4}\right)}+4\)

Step 7 ：Simplify to get \(g(f(x))=x\)

Step 8 ：Since both \(f(g(x))\) and \(g(f(x))\) simplify to \(x\), the functions \(f\) and \(g\) are inverses of each other

Step 9 ：\(\boxed{\text{Final Answer:}}\)

Step 10 ：a. \(f(g(x))=x\)

Step 11 ：b. \(g(f(x))=x\)

Step 12 ：c. \(f\) and \(g\) are inverses of each other