How do I find $f(0)$ for this function?

Question

Let $f: \mathbb R \to \mathbb R $ such that

$$(f \circ f \circ f)(x)= (f\circ f)(x)+x$$

for every $x \in \mathbb R$.

How can I compute $f(0)$?

Answer 1

The only reasonable thing to do is to plug $0$ there. As in: $$(f\circ f \circ f)(0) = (f \circ f)(0).$$ Write $f(0) = y_0$. Apply $f$ twice to get: $(f\circ f \circ f)(0) = (f \circ f)(y_0)$ and $(f \circ f)(0) = (f \circ f)(y_0)$. Applying $f$ again and using the hypothesis for $y_0$ we get: $$(f \circ f)(0) = (f \circ f \circ f)(y_0) = (f \circ f)(y_0) + y_0.$$Cancelling, we get $y_0 = f(0) = 0$.