what is the domain and range of $x^{1/3}$

Question

What is the domain and range of $x^{1/3}$?
I know what the graph looks like, but I am unsure of the domain and the range especially at $x=0$. It it all real, or is there a point of discontinuity at $x=0$?

Thanks

Answer 1

Consider the inverse function of $f(x)=x^{\frac{1}{3}}$, which is $f^{-1}(x)=x^3$.

By definition, since the domain and range of $f^{-1}(x)$ is $(-\infty, \infty)$, the domain and range of $f(x)$ is $(-\infty, \infty)$.

More importantly, because $f^{-1}(0)=0$, $f(0)=0$. This can also be observed intuitively by sketching $f^{-1}(x)=x^3$ and turning the paper sideways.