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) $f(x)=\frac{x}{3}$
\[
g(x)=3 x
\]
\[
f(g(x))=
\]
\[
g(f(x))=
\]
$f$ and $g$ are inverses of each other
$f$ and $g$ are not inverses of each other
(b) $f(x)=2 x-3$
$g(x)=\frac{x+3}{2}$
$f(g(x))=$
$g(f(x))=$
$f$ and $g$ are inverses of each other
$f$ and $g$ are not inverses of each other

Step 1 ：Let's start with the first pair of functions: \(f(x)=\frac{x}{3}\) and \(g(x)=3x\).

Step 2 ：Find the composition \(f(g(x))\), which simplifies to \(\frac{3x}{3} = x\).

Step 3 ：Find the composition \(g(f(x))\), which simplifies to \(3\left(\frac{x}{3}\right) = x\).

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

Step 5 ：Next, let's move on to the second pair of functions: \(f(x)=2x-3\) and \(g(x)=\frac{x+3}{2}\).

Step 6 ：Find the composition \(f(g(x))\), which simplifies to \(2\left(\frac{x+3}{2}\right) - 3 = x\).

Step 7 ：Find the composition \(g(f(x))\), which simplifies to \(\frac{2x-3+3}{2} = x\).

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

Step 9 ：Final Answer: \(\boxed{\text{For the first pair of functions, } f(x)=\frac{x}{3} \text{ and } g(x)=3x, f \text{ and } g \text{ are inverses of each other.}}\)

Step 10 ：Final Answer: \(\boxed{\text{For the second pair of functions, } f(x)=2x-3 \text{ and } g(x)=\frac{x+3}{2}, f \text{ and } g \text{ are inverses of each other.}}\)