$f(x)$ is a periodic function with period $T$.

Question

Prove that if $f(x)$ is a periodic function with period T, then the function $f(ax+b)$, where $a>0$, is periodic with period $\frac{T}{a}$.

I started with, $$f[(a(x+T/a)+b]=f[(ax+b)+T]=f(ax+b).$$ And hence $f(ax+b)$ is a periodic function. But I am not able to proceed further.

Answer 1

The second part is to show that no smaller number than $T/a$ would work. The usual approach is by contradiction: assume that $k < T/a$ also works, i.e. that $f(a(x+k)+b)=f(ax+b)$ and show that this would give a smaller value for the original period than $T$.