Dedicated to my beloved

let $$f$$ be a function on real numbers and whole $$x$$ such that ,

$$f (\varphi + x)+3 = f (ix)$$

also $$f(2)=7$$

Let us define another function $$P$$ on $$i$$ and $$x$$ such that ,

$\left.\begin{matrix} &P_{x} (i)\geq P_{x}(x+i) & \\ &P_{x-1}(i)+3 \geq P_{x}(x+i)& \\ &P_2 (i)=10 & \end{matrix}\right\}$ holds true , where $$i=\phi (x)$$

Then evaluate :

$\left(P_4 (i) + f \left(\frac {\sqrt5 (\sqrt5 +1)}{2}\right)\right)$

Notations:

• $$\varphi$$ denotes the Golden ratio, $$\varphi = \dfrac{1+\sqrt5}{2}$$.

• $$\phi(\cdot)$$ denotes the Euler's totient function.

• $$x$$ and $$i$$ are whole numbers.

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