Number of functions $f : A \to A$ such that $f(f(x))=f(x)$ If $A=\left\{1,2,3,4,5\right\}$ Then

Find Number of functions $f : A \to A$ such that $f(f(x))=f(x)$

Case $1.$ if $f$ is injective then $f(f(x))=f(x)$ $\implies$ $f(x)=x$, hence there is only one injective function which is an identity function.
Case $2.$ When $f$ is many to one function
Let us assume that $f(1)=f(2)=f(3)=f(4)=f(5)=k$ Then clearly for any $k$ from the set $A$, $f(f(x))=f(x)$
hence there are five such functions.
Are there any other possibilities?

There are other possibilities.  Here is one:
1 goes to 2
2 goes to 2
3 goes to 5
4 goes to 5
5 goes to 5

Expanding on this answer: The framework for these functions is to have some number (at least one) of fixed points, and have everything else map into the set of fixed points.
