Answer
$\text{Domain: }
(-\infty,\infty)
\\\text{Range: }
[0,\infty)$
Work Step by Step
The red graph is the graph of the first piece.
The blue graph is the graph of the second piece.
Using a table of values, the graph of the given piecewise function, $
f(x)=
\begin{cases}
x^2
\text{ if }
x\lt0
\\\\
\sqrt{x}
\text{ if }
x\ge0
,\end{cases}$
is shown above.
The domain is the set of all $x$-values of the graph, while the range is the set of all $y$-values. Based on the graph above, then
\begin{array}{l}\require{cancel}
\text{Domain: }
(-\infty,\infty)
\\\text{Range: }
[0,\infty)
.\end{array}