Which function is this?

Algebra Level 4

Let \(f: \mathbb{R} \rightarrow [\frac{1}{2}, 1 ] \) and \[f\left(x+2\right)= \frac{1}{2} +\sqrt{f\left(x\right)-f\left(x\right)^2}\] Then which among the following is always true?

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