# If $f$ and $g$ are periodic functions, is $g \circ f$ periodic?

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?

• Start by showing that $(g\circ f)(x+T) = (g\circ f)(x)$ for all $x$. So then you know that $T$ is a period of $g\circ f$. – Daniel Fischer Oct 5 '14 at 11:42

## 1 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 $f(x)$.

• I think part of the point of Daniel Fischer's comment is that $g \circ f$ could have a smaller period than $f$, e.g. compare the periods of $\sin x$ and $|\sin x|$. (Replace the absolute value function with a periodic function that agrees with it on $[-1,1]$ if you also want $g$ to be periodic.) – Zoe H Oct 5 '14 at 12:06
• @ Zoe: You are right that $T$ is the period of $h(x)$ as defined above but not necessarily the smallest one. – Arian Oct 5 '14 at 13:37