How to solve this function equation or plot its numerical image

I can solve the function equation $f(f(x))=x$ with the following code:

WolframAlpha["solve f(f(x))=x", {{"SolutionAsAFunctionalEquation", 1},
   "Content"}]

But for the function equation $f(f(mathrm{e}^{x}))=x^{x}$, the above code can’t solve it:

WolframAlpha["solve f(f(E^x))=x^x", {{"SolutionAsAFunctionalEquation",
    1}, "Content"}]

What can I do to solve this function equation with MMA?

In addition, if this function equation can’t get an analytic solution, can we plot the image of its numerical solution with MMA?

If none of the above can be obtained, it is acceptable to find an analytical or numerical solution with a small error in a certain interval.

In addition, there is a graphs of some interval of the function equation in the textbook. I want to know how he drew them:

Other examples for testing(cvgmt):

$forall x neq 0$, there exists $f(x)+fleft(frac{x-1}{x}right)=2 x$. Now I want to find the possible expression of $f(x)$.