Answer
$f^{-1}(x)=(x-1)^3$
$f(x)$: $(-\infty,\infty)$ and $(-\infty,\infty)$.
$f^{-1}(x)$: $(-\infty,\infty)$ and $(-\infty,\infty)$.
Work Step by Step
1. $f(x)=x^{1/3}+1 \Longrightarrow y=x^{1/3}+1 \Longrightarrow x=y^{1/3}+1 \Longrightarrow y=(x-1)^3 \Longrightarrow f^{-1}(x)=(x-1)^3$
2. check $f(f^{-1}(x))=((x-1)^3)^{1/3}+1=x$. $f^{-1}(f(x))=((x^{1/3}+1)-1)^3=x$.
3. The domain and the range of $f(x)$: $(-\infty,\infty)$ and $(-\infty,\infty)$.
The domain and the range of $f^{-1}(x)$: $(-\infty,\infty)$ and $(-\infty,\infty)$.