Motivation was that all the functional equations that appear in Olympiad have mainly one solution. for example, $xf(x^2)f(f(y))+f(yf(x))=f(xy)(f(f(x^2))+f(f(y^2)))$
You can find infinitely many of them in AOPS site.
Let's consider following functional equations.
this clearly has infinitely many non-trivial solutions, that is, $f(x)=x^a$
My question is, does my following conjecture hold?
Conjecture: we call functional equation GOOD if $f(x)=x$ is a solution, and it is consist of algebraic operation of $f$. If there are finitely many non-constant solutions. Then the equation's only non-trivial solution is $f(x)=x$.
My question could be unclear, so basically I'm asking the following.
Can you make a functional equation that only $f(x)=x, f(x)=x^2$ is solution?
More generally, is it possible to construct a functional equation that the solution set is given set?