Is $f^2 \circ f=f \circ f^2$ true?

Question

Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is a function from the set of real numbers to the same set with $f(x)=x+1$.

We write $f^{2}$ to represent $f \circ f and f^{n+1}=f^n \circ f$.

Is it true that $f^2 \circ f = f \circ f^2$?

Why?

Answer 1

Since composition of functions is associative: $$ f\circ (g \circ h) = (f \circ g) \circ h $$ then one may write $$ f^2 \circ f = (f\circ f)\circ f = f\circ (f \circ f) = f\circ f^2. $$