Question

$f(x+p)=f\left(x\right)\forall xϵ$ X if c. $xϵX$
$f(cx+p)=f\left(c(x+\frac{p}{c})\right)=f\left(cx\right)$

Answer 1

Function f(x) has a period p. Therefore f(x+p) = f(x)
For a periodic function f(x) with period p, the function f(cx) will be periodic with a period p/mod(c).
So we can say that the period of f(cx) will be $\frac{p}{c}$
So, for a function f(cx+p),
$f(cx+p)=f\left(c(x+\frac{p}{c})\right)$ (Taking p common)
Since period of f(cx) is $\frac{p}{c},f\left(c(x+\frac{p}{c})\right)=f\left(cx\right)$.