Periodicity of a function

Question

If $f(x)$ is periodic with period $a$, would $f(tx)$ be periodic with period $a/t$? Would $f(tx+b)$ make still have period $a/t$? I'm inclined to think so, because this works for the trig functions, but i'm not sure how to generalize the result to all functions. Also if $h(x)$ is a function, then would $h$ composed with $f(x)$ still be periodic with period $a$? Does periodicity change if $f$ is composed with $h$ instead?

Answer 1

Yes, $f(tx)$ is periodic with period $\frac{a}{t}$. To prove this, just note that

$$f\left(t(x + \frac{a}{t})\right) = f(tx + a) = f(tx)$$

as desired.

On the other hand, $f(x + b)$ is still periodic with period $a$; for we have

$$f\Big((x + a) + b\Big) = f\Big((x + b) + a\Big) = f(x + b)$$

Combining these two, and using a similar proof for the function $h \circ f$ can answer several more of your questions.

On the other hand, $f \circ h$ need not be periodic. Take as an example $f(x) = \sin{x}$ and $h(x) = x^2$.