In the book Putnam & Beyond, it is proven (exercise 383) that there is no non-constant function $f:(1,+\infty)\to\mathbb{R}$ such that $f(x)=f\left(\frac{1+x^2}{2}\right),\forall x>1$ and $\lim_\limits{x\to\infty}{f(x)}$ exists.
Clearly, if we remove the constraint that $\lim_\limits{x\to\infty}{f(x)}$ exists, there exist infinitely many discontinuous functions that satisfy the rest of the hypothesis. So, the question here would be for continuous functions (and obviously $\lim_\limits{x\to\infty}{f(x)}$ shouldn't exist as is trivially proven from the above exercise). Therefore, let's state the modified version of the problem:
Is there any non-constant, continuous function $f:(1,+\infty)\to\mathbb{R}$ such that $$f(x)=f\left(\frac{1+x^2}{2}\right),\forall x>1$$ and if yes, is it expressible with elementary functions or not?
Note: I believe it all comes down to the density (or not) within the real numbers of the sequence: $x_0$:arbitrary and $$x_{n+1}=\frac{1+x_n^2}{2},\forall n\geq0$$