Thread: Periodic functions

If $\displaystyle f(a-x)=f(a+x);f(b-x)=f(b+x),\forall x\in R, a>b$.Then prove that the function $\displaystyle f(x)$ is periodic and hence find its period.

Originally Posted by pankaj

If $\displaystyle f(a-x)=f(a+x);f(b-x)=f(b+x),\forall x\in R, a>b$.Then prove that the function $\displaystyle f(x)$ is periodic and hence find its period.

Let $\displaystyle x = a - t$ so the first gives $\displaystyle f(t) = f(2a - t)$

Let $\displaystyle x = b - t$ so the secondgives $\displaystyle f(t) = f(2b-t)$

Equating gives

Let $\displaystyle f(2a-t) = f(2b-t)$ and setting $\displaystyle T=2a-t$ first gives $\displaystyle f(T) = f(T+2b-2a)$ and so the period is $\displaystyle 2b-2a$