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) \[ \begin{array}{l} f(x)=\frac{1}{3 x}, x \neq 0 \\ g(x)=\frac{1}{3 x}, x \neq 0 \\ f(g(x))=\square \\ g(f(x))=\square \end{array} \] $f$ and $g$ are inverses of each other $f$ and $g$ are not inverses of each other. (b) $f(x)=x+4$ \[ \begin{array}{l} g(x)=x+4 \\ f(g(x))=\square \\ g(f(x))=\square \end{array} \] $f$ and $g$ are inverses of each other $f$ and $g$ are not inverses of each other

Step 1 ：First, we need to find f(g(x)) and g(f(x)).

Step 2 ：For f(g(x)), we substitute g(x) into f(x), so we get: \(f(g(x)) = f(\frac{1}{3x}) = \frac{1}{3*(\frac{1}{3x})} = x\)

Step 3 ：For g(f(x)), we substitute f(x) into g(x), so we get: \(g(f(x)) = g(\frac{1}{3x}) = \frac{1}{3*(\frac{1}{3x})} = x\)

Step 4 ：Since \(f(g(x)) = g(f(x)) = x\), f and g are inverses of each other. \(\boxed{f \text{ and } g \text{ are inverses of each other}}\)

Step 5 ：For f(g(x)), we substitute g(x) into f(x), so we get: \(f(g(x)) = f(x+4) = (x+4) + 4 = x + 8\)

Step 6 ：For g(f(x)), we substitute f(x) into g(x), so we get: \(g(f(x)) = g(x+4) = (x+4) + 4 = x + 8\)

Step 7 ：Since \(f(g(x)) = g(f(x)) = x + 8\), which is not equal to x, f and g are not inverses of each other. \(\boxed{f \text{ and } g \text{ are not inverses of each other}}\)