math-combo-01

# Counting Idempotent Self-Maps

Let $A = \{1, 2, 3, 4, 5\}$. Count the number of functions $f : A \to A$ that satisfy
$$f(f(x)) = f(x) \qquad \text{for every } x \in A.$$

(Such a function acts as the identity on its own range.)

Give your answer as a single integer.

To find the number of functions $f: A \to A$ satisfying $f(f(x)) = f(x)$ for all $x \in A$, we analyze the properties of such idempotent functions.

### 1. Characterization of Idempotent Functions
The condition $f(f(x)) = f(x)$ implies that $f$ acts as the identity on its range $R = f(A)$.
*   If $y \in R$, then $y = f(x)$ for some $x$. Thus, $f(y) = f(f(x)) = f(x) = y$.
*   Conversely, if $f(y) = y$ for all $y \in R