For each pair of functions $f$ and $g$ below, find $f(g(x))$ and $g(f(x))$. Then, determine whether $f$ and $g$ are inverses of each other.
Simplify your answers as much as possible.
(Assume that your expressions are defined for all $x$ in the domain of the composition. You do not have to indicate the domain.)
(a)
\[
\begin{array}{l}
f(x)=\frac{3}{x}, x \neq 0 \\
g(x)=\frac{3}{x}, x \neq 0 \\
f(g(x))=\square \\
g(f(x))=\square
\end{array}
\]
$f$ and $g$ are inverses of each other
$f$ and $g$ are not inverses of each other
(b) $f(x)=x+4$
\[
g(x)=-x+4
\]
\[
f(g(x))=
\]
\[
g(f(x))=
\]
$f$ and $g$ are inverses of each other
$f$ and $g$ are not inverses of each other

Step 1 ：Given the functions \(f(x) = \frac{3}{x}\) and \(g(x) = \frac{3}{x}\), we need to find \(f(g(x))\) and \(g(f(x))\).

Step 2 ：Substitute \(g(x)\) into \(f(x)\) to get \(f(g(x)) = \frac{3}{\frac{3}{x}} = x\).

Step 3 ：Substitute \(f(x)\) into \(g(x)\) to get \(g(f(x)) = \frac{3}{\frac{3}{x}} = x\).

Step 4 ：Since both \(f(g(x))\) and \(g(f(x))\) equal to \(x\), \(f\) and \(g\) are inverses of each other.

Step 5 ：Given the functions \(f(x) = x + 4\) and \(g(x) = 4 - x\), we need to find \(f(g(x))\) and \(g(f(x))\).

Step 6 ：Substitute \(g(x)\) into \(f(x)\) to get \(f(g(x)) = (4 - x) + 4 = 8 - x\).

Step 7 ：Substitute \(f(x)\) into \(g(x)\) to get \(g(f(x)) = 4 - (x + 4) = -x\).

Step 8 ：Since neither \(f(g(x))\) nor \(g(f(x))\) equal to \(x\), \(f\) and \(g\) are not inverses of each other.

Step 9 ：\(\boxed{\text{For the first pair of functions, } f \text{ and } g \text{ are inverses of each other. For the second pair of functions, } f \text{ and } g \text{ are not inverses of each other.}}\)