Show that if $f(x)$ is a periodic function with period $P$, then $g(x)=f(kx)$ is also periodic and define a period of the periodic function $g(x)$. Afterwards, find a periodic function with the period of $1$.

I need this for my uni and I'm kinda stuck.

  • 2
    $\begingroup$ Hint: $g(x+a)=f\left(k\times(x+a)\right)=f\left(kx+ka\right)$. Does that suggest a good candidate for $a$? $\endgroup$ – lulu Oct 31 '16 at 18:07
  • $\begingroup$ This may attract negative attention because it looks a bit like a verbatim assignment. To dispel such thoughts you could, for example, review a few facts about trig functions and share your thoughts about the periodicity of $g(x)=\sin 2x$. IOW, what seems to happen with $P=\pi, k=2$? $\endgroup$ – Jyrki Lahtonen Oct 31 '16 at 21:08

If $f(x)$ has period $P$, then $f(x) = f(x+P)$ for all $x$. Then look at the function $g$. What is the value of $g(x + \frac{P}{k})$? How can you write this in terms of $f$?

  • $\begingroup$ What? I don't really get it. Will it be f(kx+p/k)? $\endgroup$ – George K. Oct 31 '16 at 19:00
  • $\begingroup$ Note quite. Remember $g(x) = f(kx)$ - this is true no matter what $x$ is - you need to multiply the entire argument by $k$. For example, $g(a+b) = f(k(a+b)) = f(ka+kb)$, NOT $f(ka+b)$. So think about $g(x + \frac{P}{k}) = f(k(x + \frac{P}{k}))$. $\endgroup$ – Nitin Oct 31 '16 at 19:23
  • $\begingroup$ so should I write f(kx) = f(kx + P) = f(k(x+P/k)) ? and how does that prove that g(x) is periodic. Sorry if it's obvious but I struggle a lot with this one! $\endgroup$ – George K. Oct 31 '16 at 21:40
  • $\begingroup$ Very good. Now, as you wrote $f(kx) = f(k(x+\frac{P}{k}))$. Can you now write these in terms of $g$? $\endgroup$ – Nitin Oct 31 '16 at 22:25

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