#### Answer

$f$ and $g$ are inverses of each other.

#### Work Step by Step

RECALL:
(1) $(f \circ g)(x) = f\left(g(x)\right)$
(2) The function $g(x)$ is the inverse of function of a one-to-one function $(x)$ if for every element of the domain,
$(f \circ g)(x) =x$ and $(g \circ f)(x)=x$
Find $(f \circ g)(x)$ by substituting $\frac{1}{3}x-3$ to $x$ in $f(x)$:
$(f\circ g)(x)
\\=f\left(g(x)\right)
\\=3\left(\frac{1}{3}x-3\right)+9
\\=3(\frac{1}{3}x)-3(3)+9
\\=x-9+9
\\=x$
Find $(g \circ f)(x)$ by substituting $3x+9$ to $x$ in $g(x)$:
$(g\circ f)(x)
\\=g(\left(f(x)\right)
\\=\frac{1}{3}\left(3x+9\right)-3
\\=\frac{1}{3}(3x) + \frac{1}{3}(3)-3
\\=x+3-3
\\=x$
Since $(f\circ g)(x) = (g\circ f)(x)=x$, then $f$ and $g$ are inverses of each other.