Answer
$(f \circ g)(x)=x$
$(g \circ f)(x)=x$
$f$ and $g$ are inverses of each other.
Work Step by Step
Since $f(x)=\sqrt[3]{7 x+5}, \ g(x)=\frac{x^{3}-5}{7}$, we have
$$\begin{align*}
(f \circ g)(x)&=f(g(x))\\
&=f\left(\frac{x^{3}-5}{7}\right)\\
&=\sqrt[3]{7\left(\frac{x^{3}-5}{7}\right)+5}\\
&=\sqrt[3]{ x^3}\\
&=x
\end{align*}$$
and
$$\begin{align*}
(g \circ f)(x)&=g(f(x))\\
&=g(\sqrt[3]{7 x+5})\\
&=\frac{(\sqrt[3]{7 x+5})^{3}-5}{7}\\
&=\frac{({7 x+5})-5}{7}\\
&=x .
\end{align*}$$
Since $(f \circ g)(x) = x$ and $(g \circ f)(x) = x$, then $f$ and $g$ are inverses of each other.