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 :

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

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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.