Given two piece-wise functions, prove their composition is periodic.

Question

I know how to graph h(f(x)), however, I am having trouble proving it is periodic. i. We must prove that h(f(x + k)) = h(f(x)) for all x ∈ ℝ and for some k ≠0 ∈ ℝ in order to prove that h(f(x)) is periodic. I think I have found general rules for f(x) and h(x) but I have defined them in a very wordy way.

f(x) = x – n if n ≤ x < n + 2 where n is the first even integer that is less than x ∈ ℝ. [For example, if x = 1.2, then n = 0 and n + 2 = 2.]

h(x) = 2x – p if p ≤ x < p + 1 where p is the first integer that is less than x ∈ ℝ. [For example, if x = -2.2, then p = -3 and p + 1 = -2].

However, if I try to algebraically compose this it gets messy, and I don't think I'm headed in the right direction to prove that it's periodic. Any suggestions?

Answer 1

If $T$ is a period of $f$, then $T$ is a period of $g\circ f$.

Since $f(x+T)=f(x) \forall x\in\mathbb{R}$, then $g\circ f(x+T)=g(f(x+T))=g(f(x))=g\circ f(x)$. In your case, set $T=2$.

Answer 2

A periodic function $a(x)$ satisfies $a(x)=a(x+k)$ for some $k$ and all $x$. $f(x)$ clearly satisfies this in this instance ($k=2$) and is therefore periodic. Therefore, if there exist $m,d$ such that $h(x+m)=h(x)+d$ for all $x$, then $f\circ h$ will be periodic (with period LCM($d,k$), as a matter of fact). $h(x)$ satisfies this ($m=3$) and therefore $f \circ h$ is periodic.