I was doing a few proofs involving iterated functions such as the proofs that
If $()$ is injective, then $f^n(x)$ is injective.
If $()$ is not injective, then $f^n(x)$ is not injective.
If $()$ is symmetric about the line $x=c$, then $f^n(x)$ is as well, for positive integral $n$.
The third statement is what is frustrating me, because it is easily proven. However, the converse seems to be true as well, but it isn't so easy to prove. Can anybody help? If it would be helpful for me to type up the proofs for the first three, I'd be glad to, but they're all pretty self-explanatory.
The converse to (3) is false. For example, take $n=2$ and $$ f(x) = |x|-x = \begin{cases} 2|x|, &\text{if } x<0, \\ 0, &\text{if } x\ge0. \end{cases}$$