Answer
Since $f\circ g(x)=x$ and $g\circ f(x)=x $
f and g are inverses of each other.
Work Step by Step
$\begin{array}{lllll}
f\circ g(x) & =f[g(x)] & ..... & g\circ f(x) & =g[f(x)]\\
& =[g(x)]^{3}+1 & & & =[f(x)-1]^{1/3}\\
& =[(x-1)^{1/3}]^{3}+1 & & & =[x^{3}+1-1]^{1/3}\\
& =(x-1)+1 & & & =(x^{3})^{1/3}\\
& =x & & & =x
\end{array}$
Since $f\circ g(x)=x$ and $g\circ f(x)=x $
f and g are inverses of each other.