What is the domain of f(x)=5x3−(x−2)?

That numerator doesn't impose any restrictions, but there are two constraints from the denominator: the whole denominator cannot be 0 and the number inside the radical cannot be negative. Let's look at each in turn.
The denominator cannot be 0 So then we cannot have ${\displaystyle 3-\sqrt{x}-2=0}$. So let's find out which x's make this true so that we can exclude them from the domain.
${\displaystyle 3=\sqrt{x-2}}$
$9=x-2$
$11=x$
So 11 cannot be in the domain.
The number under the radical cannot be negative. That is, we must have ${\displaystyle x-2\ge 0}$. Hence ${\displaystyle x\ge 2}$.
Conclusion
Putting these two conditions together we get that the domain is ${\displaystyle [2,\infty )}$ without the number 11. Hence
Domain ${\displaystyle \left(f\right)=[2,11)\cup (11,\infty )}$