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.

Illustration:  Find the period of the following functions

(i) f(x) = sinx + {x}   

(ii) f(x) = tan(x/3) + sin 2x.

(iii) f(x) = |sinx| + |cosx|      

(iv) f(x) = ((1+sin x)(1+sec x))/((1+cos x)(1+cosec x))

Solution:

(i) Here f(x) = sinx + {x}

Period of sinx is 2p and that of {x} is 1. But the L.C.M. of 2p and 1 does not exist. Hence sinx + {x} is not periodic.

(ii) Here f(x) = tanx/3 + sin2x. Here tan(x/3). Here tan(x/3) is periodic with period 3p and sin2x is periodic with period p. 

Hence f(x) will be periodic with period 3p.

(iii) Here f(x) = |sinx| + |cosx|