The function $ \psi : \mathbb R \to \mathbb R $ satisfies the relation: $$ \psi ( x + 2 ) = 1 + \sqrt{ 2 \psi ( x ) - \psi ( x ) ^ 2 } \text , $$ for all real $ x $. What properties does such function have? Give an example of of such function that is not constant.
My work so far:
First of all, note that the expression under the radical must be always nonnegative if we want the equation to make sense for all $ x \in \mathbb R $. So, the question should ask to find all $ \psi : \mathbb R \to [ 0 , 2 ] $ satisfying $$ \psi ( x + 2 ) = 1 + \sqrt { 2 \psi ( x ) - \psi ( x ) ^ 2 } \tag 0 \label 0 $$ for all $ x \in \mathbb R $. Consider such $ \psi $. Substituting $ x - 2 $ for $ x $ in \eqref{0} shows that one must have $ \psi ( x ) \ge 1 $ for all $ x \in \mathbb R $. Define $ \phi : \mathbb R \to \left [ - \frac 1 2 , \frac 1 2 \right ] $ with $ \phi ( x ) = \bigl ( \psi ( x ) - 1 \bigr ) ^ 2 - \frac 1 2 $ for all $ x \in \mathbb R $. Then subtracting $ 1 $ from both sides of \eqref{0} and squaring both sides of the obtained equation, we get $$ \phi ( x + 2 ) = - \phi ( x ) \tag 1 \label 1 $$ for all $ x \in \mathbb R $. Conversely, if you are given a $ \phi : \mathbb R \to \left [ - \frac 1 2 , \frac 1 2 \right ] $ satisfying \eqref{1} for all $ x \in \mathbb R $, you can define $ \psi : \mathbb R \to [ 1 , 2 ] $ with $ \psi ( x ) = 1 + \sqrt { \phi ( x ) + \frac 1 2 } $ for all $ x \in \mathbb R $, and verify that \eqref{0} holds for all $ x \in \mathbb R $. Thus $ \psi $ is a solution to our problem iff it's of the mentioned form for some $ \phi : \mathbb R \to \left [ - \frac 1 2 , \frac 1 2 \right ] $.
If you're not yet satisfied with this characterization, we can analyze such $ \phi $ a little bit more. \eqref{1} shows that $ \phi $ is uniquely determined by its values on a given interval of length $ 2 $. For example, we can define $ \eta : [ 0 , 2 ) \to \left [ - \frac 1 2 , \frac 1 2 \right ] $ as the restriction of $ \phi $ to the domain $ [ 0 , 2 ) $. Then, one can use \eqref{1} and mathematical induction on the absolute value of the integer $ n $ to verify that $$ \phi ( x + 2 n ) = ( - 1 ) ^ n \eta ( x ) $$ for all $ x \in [ 0 , 2 ) $ and all $ n \in \mathbb Z $. Conversely, given any function $ \eta : [ 0 , 2 ) \to \left [ - \frac 1 2 , \frac 1 2 \right ] $, if you define $ \phi : \mathbb R \to \left [ - \frac 1 2 , \frac 1 2 \right ] $ with $$ \phi ( x ) = ( - 1 ) ^ { \left \lfloor \frac x 2 \right \rfloor } \eta \left ( x - 2 \left \lfloor \frac x 2 \right \rfloor \right ) $$ for all $ x \in \mathbb R $, then \eqref{1} will hold for all $ x \in \mathbb R $. Therefore, we get a characterization of those $ \phi $ that are suitable for our problem. Note that any $ \eta $ with the mentioned domain and codomain will work, and thus we can't go any further.
In case you need $ \psi $ to satisfy further regularity conditions like continuity, differentiability, smoothness (in the sense of being $ n $ times continuously differentiable for some positive integer $ n $, or even being infinitely differentiable), etc., the corresponding condition on $ \eta $ must be taken into account: $ \eta $ itself must be regular in the same sense, and some compatibility conditions regarding the behavior of $ \eta $ at $ 0 $ and near $ 2 $ must hold (for example, $ ( - 1 ) ^ n \eta ( 0 ) = \lim _ { x \to 2 ^ - } \eta ^ { ( n ) } ( x ) $, where $ \eta ^ { ( n ) } $ stands for the $ n $-th order derivative of $ \eta $). Again, there will be uncountably many $ \eta $ satisfying those conditions, and we can't go any further in characterizing such functions.
A somewhat more interesting case happens when you ask for real analytic $ \phi : \mathbb R \to \left [ - \frac 1 2 , \frac 1 2 \right ] $ satisfying \eqref{1} for all $ x \in \mathbb R $. While, again, there will be uncountably many suitable $ \eta $, you will get nice representations of $ \phi $ in terms of Fourier series. For suitable conditions on $ \eta $ for our purpose, see "Do all analytic and $ 2 \pi $ periodic functions have a finite Fourier series?", and for extending such $ \eta $ to the whole real line (and in fact, whole complex plane) to get $ \phi $, see "Prove that periodic analytic function can be written as $ \sum _ { - \infty } ^ \infty c _ n e ^ { 2 \pi i n z } $".
Finally, whether you consider further regularity conditions on $ \psi $ or not, you should investigate the properties of $ \phi $ to see what properties $ \psi $ has. For example, the fact that $ \psi $ is $ 4 $-periodic (mentioned in your attempt) comes from $ 4 $-periodicity of $ \phi $, which is a consequence of applying \eqref{1} twice. Essentially, \eqref{1} is all you get in the general case, and every other property will be a consequence of that.