Let $f(x)$ be a periodic function with period $p$. Then $$f(x+p)=f(x)$$ Let, $$g(x)=f(ax+b)+c$$ Now it's mentioned in my book that $g$ is also a periodic function with a period of $\frac{p}{|a|}$.
My questions:-
$(1)$ Why does period have $|a|$ in it's denomination and why not $a$ simply?
My understanding:- I am saying that it should be $a$ because, $$g(x+\frac{p}{a})=f(ax+p+b)+c$$ Since $f$ is periodic and have a period of $p$ then, $$g(x+\frac{p}{a})=g(x)$$ And hence $g$ has a period of $\frac{p}{a}$.
Then why is that modulus sign ?
The Period of a real valued function is a positive value by definition. So $|a|$ is written instead of $a$ to guarantee that you calculate a positive number for $p/|a|$.