Answer
$\left[ 1,\infty \right)$
Work Step by Step
The radicand of the given function, $
f(x)=\sqrt{x^3-1}
,$ can only be a nonnegative number. Hence,
\begin{array}{l}\require{cancel}
x^3-1\ge0
\\\\
x^3\ge1
\\\\
x\ge\sqrt[3]{1}
\\\\
x\ge1
.\end{array}
Hence, the domain is $
\left[ 1,\infty \right)
.$
