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

Step 1 ：Given the first pair of functions, $f(x) = \frac{x}{3}$ and $g(x) = 3x$, we substitute $g(x)$ into $f(x)$ to get $f(g(x))$ and vice versa to get $g(f(x))$.

Step 2 ：Calculating $f(g(x))$, we substitute $g(x)$ into $f(x)$: $f(g(x)) = f(3x) = \frac{3x}{3} = x$.

Step 3 ：Calculating $g(f(x))$, we substitute $f(x)$ into $g(x)$: $g(f(x)) = g(\frac{x}{3}) = 3\cdot\frac{x}{3} = x$.

Step 4 ：Since both $f(g(x))$ and $g(f(x))$ equal to $x$, $f$ and $g$ are inverses of each other for the first pair of functions.

Step 5 ：Given the second pair of functions, $f(x) = 2x - 3$ and $g(x) = \frac{x+3}{2}$, we substitute $g(x)$ into $f(x)$ to get $f(g(x))$ and vice versa to get $g(f(x))$.

Step 6 ：Calculating $f(g(x))$, we substitute $g(x)$ into $f(x)$: $f(g(x)) = f(\frac{x+3}{2}) = 2\cdot\frac{x+3}{2} - 3 = x$.

Step 7 ：Calculating $g(f(x))$, we substitute $f(x)$ into $g(x)$: $g(f(x)) = g(2x-3) = \frac{2x-3+3}{2} = x$.

Step 8 ：Since both $f(g(x))$ and $g(f(x))$ equal to $x$, $f$ and $g$ are inverses of each other for the second pair of functions.

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

Step 10 ：\(\boxed{\text{For the second pair of functions, } f \text{ and } g \text{ are inverses of each other.}}\)