Domain: we have a square root. A square root only accepts as input a non-negative number. So we have to ask ourselves: when is x^3 \ge 0? It's easy to observe that, if x is positive, then x^3 is positive too; if x=0 then of course x^3=0, and if x is negative, then x^3 is negative, too. So, the domain (which, again, is the set of numbers such that x^3 is positive or zero) is [0,\infty).
Range: now we have to ask which values the function can assume. The square root of a number is, by definition, not negative. So, the range can't go below 0? Is 0 included? This question is equivalent to: is there a value x such that sqrt(x^3)=0? This happens if and only if there is an x value such that x^3=0, and we've already seen that the value exists and is x=0. So, the range starts from 0. How further does it go?
We can observe that, as x gets large, x^3 get even larger, growing to infinity. Same goes for the square root: if a number gets larger and larger, so does its square root. So, sqrt(x^3) is a combination of quantities which grow boundless to infinity, and thus the range has no bounds.
