Solving for $f$

Question

How should i solve expressions in $f$?

$$ f(f(x))=e^x $$

or:

$$ f(f(x+1)+1)=x $$

Series, expansions, discretizations, CAS, whatever?

My normal approach is useless here...

Answer 1

I don't know how to solve the first one, but just by guessing, one solution to the it is $f(x) = x-1$, so $f(x+1) = x$, $f(x+1)+1 = x+1$, $f(f(x+1)+1) = x$