What is the domain of - 3x^ (1/3) + 4x^ (1/2)

The domain of the function would be [0, ∞)

This is because of the x^ (1/2); this is the same as the square root of x. Remember: if x is under the square root, then x can't be negative because negatives do not have square roots. X can be 0, because the square root of 0 is 0, and of course it can be positive. So, x can be anywhere from 0 to infinity including 0.