Let $f$ be a $2\pi$ periodic function defined as $$ f=|x|=\begin{cases} -x&x\in[-\pi,0[\\ x&x\in[0,\pi[ \end{cases} $$
What is $g(x)=f(2x)$ and $h(x)=f(x)+\frac{f(2x)}{2}$?
Using the definition of $f$ I get that $$ \begin{split} g(x)&=|2x|=\begin{cases} -2x&x\in[-\pi,0[\\ 2x&x\in[0,\pi[ \end{cases} \\ h(x)&= |x|+\frac{g(x)}{2}= \begin{cases} -x-x&x\in[-\pi,0[\\ x+x&x\in[0,\pi[ \end{cases} = g(x) \end{split} $$ I'm quite sure this is wrong - can anyone help em out?