I am having trouble understanding the $x^e$ function. Indeed, I cannot draw it with my software, it only offers me a graph on $R^+$. This function is however defined on any $R$?
Is this because $e$ is irrational?
I tried to understand why, I thought maybe if we consider that $x$ as $e^{\log(x)}$, then it must not work for $x<0$? but when I do $e^{-12}$ I get a real value of $6.14421\times10^{-6}$
The calculation $x^e$ does not produce real values when $x<0$. (There is a problem even with $(-1)^{1/2}$, which you may already have seen.) With some effort, we can interpret $x^e$ as a complex multi-valued function.
See mathworld