What is the period of $f(2x)$ if $f(2x+6) = f(2x)$ and that of $f(x)$?

Question

The question is in the title. Also note that $f(x)$ is non-constant function.

This is not same as other question asked from similar title.

I understand that if I put $x+3$ in place of $x$ in $f(2x)$, I get the same function. So the period must be $3$. And that the period of $f(x)$ must be $6$.

But 'The book' says period of $f(2x)$ is $6$ and of $f(x)$ is $12$. I cannot understand this.

Answer 1

You are correct, in finding the fundamental period of the function.

Therefore, if you have found the fundamental period of $f(x)$ is $6$, there is no harm in saying that it has a period of $12$, too.

_{(Although I understand that when nothing is specified, then we assume that the question is about fundamental period, but if it's a MCQ question and you have no other option that is correct, than this must be the most correct option.)}

Answer 2

The function $f(x)$ clearly has period $6$ because you have an equation of the form $f(y)=f(y+6)$ - whether this is the smallest, or fundamental, period of the function depends on the wording of the question.

For $f(2x)$ - since there seem to have been one or two confused comments - it is perhaps as well to study $g(x)=f(2x)$ and determine the period of $g(x)$. We have, as you have noted $$g(x+3)=f(2x+6)=f(2x)=g(x)$$ and the period is $3$ (or $r=3/n$ for $n$ a positive integer).