As we know from 1st year Calculus, for function $f(x) = x^n, x \in \mathbb{R}, n \in \mathbb{N}$, if $n$ is odd, the $f(x)$ is increasing; if $n$ is even $f(x)$ is increasing when $x \geq 0$ and decreasing when $x<0$. But here we only have the cases where $n$ is a positive integer. We don't know how it goes if $n$ is replaced by some irrational number like $\pi$ or $e$.
So I tried to do the plot of $y_1 = x^e$ and $y_2 = x^{\pi}$ through MATLAB to study their monotonicity, and here are the codes and plots of them:
$\textbf{Codes}:$
$\textbf{Plots}:$
And from the plots above, we can see that $y_1 = x^e$ and $y_2 = x^{\pi}$ are monotonically increasing on $\mathbb{R}$, and I am quite curious about how to rigorously prove the monotonicity of them. Can somebody give me some hint on which mathematical tool I should use here, for example derivative or what? Thanks!
As you can see there is a warning given which says that the imaginary parts are ignored.
It is actually trying to say that in the case of functions like these we cannot have $x<0$ because then we will get an imaginary number. (Recall that $\textrm{(-ve number)}^{\textrm{irrational}}$ yeilds an imaginary number.)
Now let's say that you define $f:\mathbb{R}\to\mathbb{C}$ and now you want to study the nature of the curve when $x<0$. Then this is actually absurd as there is no such thing as "larger number" in complex numbers.
Suppose $x<0$ then let $x=-a$ where $a>0$. Now $$(-a)^{e}=(-1)^ea^e$$ $$=(e^{i\pi})^e a^e=a^e\cdot e^{i\pi e}$$ $$=a^e(\cos(\pi e)+i\sin(\pi e))$$ which is indeed a complex number.