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.