Period of a periodic function

Question

I have a exercise where I have a given function $f(x)$ with period $p$. I need to show that $f(kx)$ has period $p/k$ and $f(x/k)$ has period $kp$ for $k>0$. I also need to show that $f(kx)$ has also period $p$ for $k>0$. I can see that it is correct, but what steps do I have to do?

I know that $f(x)=f(x+p)$, then $f(kx)=f(kx+p)$...but I do not feel that this leads to a proof.

Thanks!

Answer 1

Hint: Well, let $$g(x)=f(kx)$$. Then, for every $x\in\mathbb{R}$ we have: $$g\left(x+\frac{p}{k}\right)=f\left(k\left(x+\frac{p}{k}\right)\right)=f(kx+p=f(kx)=g(x)$$

In the same way you can show the rest of the requested results.