#### 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 $\sqrt{x-3}$ to $x$ in $f(x)$:
$(f\circ g)(x)
\\=f\left(g(x)\right)
\\=(\sqrt{x-3})^2+3
\\=x-3+3
\\=x$
Find $(g \circ f)(x)$ by substituting $x^2+3$ to $x$ in $g(x)$:
$(g\circ f)(x)
\\=g(\left(f(x)\right)
\\=\sqrt{x^2+3-3}
\\=\sqrt{x^2}
\\=|x|$
Since $x \ge 0$, then
$|x|=x$
Since $(f\circ g)(x) = (g\circ f)(x)=x$, then $f$ and $g$ are inverses of each other.