If $f$ and $g$ are periodic functions, is $g \circ f$ periodic? If it is, what is the period?
So I know:
$f(x) = f(x + T), T \in R$
$g(x) = g(x + P), P \in R$
I have this question for my homework. I don't know how to start. Intuitively I would say that is some combination of periods of each function (T+P, T-P, or something else). Using some online graphing calculators and ploting $f(x)=tan(sin(x))$ and $f(x)=sin(tan(x))$ I came to conclusion that the period of the composition is the period of the "inner" function $f(x)$. But how to show/prove that?
Best Answer
Let $h(x)=g(f(x))$ then because $f(x)=f(x+T)$ one has $$h(x)=g(f(x))=g(f(x+T))=h(x+T)$$ Therefore $h(x)$ is periodic with the same period as $f(x)$.