Domain of f is x >= 0

You are absolutely right! This works because of the odd denominator in the fractional power. I was confused because, in general, the n-th root of a negative number has n solutions, if you allow complex numbers. This is also the reason why google & wolframalpha etc tell you that (-1)^(2/7) is a complex number. None of that fanciness is necessary here, and your solution is very clear. Good job.