Domain of a Root

A radical function has its input under a root, like f(x) = √(2x - 10). Some inputs break it. With input 1 the inside is 2 - 10, which is -8, and no real number squares to a negative. The domain is every input the rule accepts.

To find it, ask for the inside to be at least 0: 2x - 10 >= 0. Move the -10 across to get 2x >= 10, then divide by 2: x >= 5. The input 5 itself is allowed, since the square root of 0 is 0.

A cube root has no such limit. cbrt(-8) is -2, because -2 cubed is -8. So g(x) = cbrt(x - 8) accepts every real input. Only an even root, like the square root, shuts out negatives.