Fractional exponentiation of functions

Question

I run into a simple question about the fractional exponentiation of a function. Suppose that $f(x)$ is a real valued function which is always positive definite on its domain. Is it possible to obtain a negative value by a fractional exponentiation of the function $f(x)$? i.e., the fractional exponentiation ${\left( {f(x)} \right)^{\frac{m}{n}}}$ (where $m$ and $n$ are some integers) could lead to a negative result? if yes, when $m$ be an even integer, can we still obtain a negative result?

Of course, I know that this question is reminiscent of the fractional exponentiation of real numbers in calculus. Can we simply apply our knowledge about fractional exponentiation of real numbers to functions?

Thank you very much in advance.

Answer 1

It depends how you define fractional exponentiation.

For example, we typically view ${x^{\frac{1}{2}}}$ with being synonymous with "${\sqrt{x}}$" (which is always positive); but perhaps a better, more complete view is to think of ${x^{\frac{1}{2}}}$ as being a set of values satisfying the fact that their square gives ${x}$: $$ x^{\frac{1}{2}} = \{a\ |\ a^2 = a\times a = x\} $$ in this way, ${\sqrt{x}}$ (which always gives a positive value) is just one of the solutions in ${x^{\frac{1}{2}}}$. In fact, we have that $$ \sqrt{x},-\sqrt{x} \in x^{\frac{1}{2}} $$ So just take ${\left(f(x)\right)^{\frac{1}{2}}}$ as the negative solution and you are set.

I WANT TO MAKE IT CLEAR THOUGH: if you are treating fractional exponentiation as being synonymous with only taking the positive roots, then no, you will obviously never get a negative result. Hopefully this helps.

Answer 2

You are only exponentiating the function values when it comes down to it, and these are real numbers, so of course what you learned in calculus applies. No, you'll never get negative values.

Answer 3

Your question is not specific to functions, but essentially to exponentiation of positive reals. Which always yields a positive result.

Anyway, as pointed by @Riemann'sPointyNose, there can be a confusion with the roots of a real, which can to some extent be negative (for a rational with even denominator).