What is the domain of x^(1/3)-x^(2/3)? I graphed it with different programs, and some said it has to be only positive real numbers, but I don't see why.
Hey Imbilio.
In terms of the domain, this should be defined by all values of x and give a real answer and you can check by using Eulers formula to verify all values are all numbers.
x^(1/3) = CUBE(x) which is defined for all x since y^3 = x and y is unique (I'm thinking in terms of inverse functions).
As for x^(2/3) we have y^(3/2) = x but y is not defined for x < 0 since SQRT(z) is not defined in the real numbers when z < 0.
But in terms of the function, y will produce a value for all values of x but you will never get an inverse function.
You will never get an inverse unless you have positive x but you should always be able to generate the function for all real values of x.