Functions

Can we Find any two Distinct Functions
$$f:\Re \rightarrow \Re$$ such that

$$f\left ( x \right )=f\left (\frac{x}{2}\right )$$

Necessary and Sufficient Conditions for the General Second Degree Equation
$$Ax^2+2Hxy+By^2+2Gx+2Fy+C=0$$
 to Represent Pair of Straight Lines:

$$\Delta =\begin{vmatrix} A &  H &G\\  H& B & F\\  G& F& C \end{vmatrix} =0$$


Case 1 :if $$H^2 > AB$$ Then they are Intersecting pair of Straight Lines

Case 2: if $$H^2 = AB$$ Then they are pair of Parallel Straight Lines

Case 3: if $$H^2 < AB$$ Then they Represent a Point in a Plane

Remainder Problem

Find the Remainder when the number $N$ is divided by 7, where

$$N=2222^{5555}+5555^{2222}$$