Periodic function
 
These are the function, whose value repeats after a fixed constant interval called period, and which makes a class of a widely used function.
        A function f of x, such that:
        f(T + x) = f(x) ∀  x ε domain of f.
        The least positive real value of T for, which above relation is true, is called the fundamental period or just the period of the function.
        e.g. for f(x) = sin x  ∀  x ε R.
We know that sin (2∏ + x) = sin x, ∀  x ε R
so f(x) = sin x is a periodic function with a period of 2∏ radians.

 
  Rules for finding the period of the periodic functions

(i)     If f(x) is periodic with period p, then a f(x) + b, where a, b ε R (a≠0) is also a periodic function with period p.
(ii)    If f(x) is periodic with period, then f(ax + b), where a ε R -{0} and b ε R, is also periodic with period p/|a|.
(iii)    let us suppose that f(x) is periodic with period p and g(x) is periodic with period q. Let r be the L.C.M. of p and q, if it exists.
(a)    If f(x) and g(x) cannot be interchanged by adding a least positive number k, then r is the period of f(x) + g(x).
(b)    If f(x) and g(x) can be interchanged by adding a least positive number k and if k < r, then k is the period of f(x) + g(x). Otherwise r is the period.