Properties of Periodic Functions
Periodic Functions: A function f (x) is said to be a periodic function if there exists a positive real number T such that f (x + T) = f (x) for all x ϵ R.
for all values of x in the domain. If there exists a least positive real number T with this property, it is called the fundamental period (also primitive period, basic period, or prime period.) A function with period T will repeat on intervals of length T and these intervals are referred to as periods.
Examples: sin (x + 2π) = sinx for all x ϵ R.
cos (x + 2π) = cosx for all x ϵ R.
Properties of periodic function:
1. If a function f is periodic with period P, then for all x in the domain of f and all positive integer n.
f (x + nP) = f (x)
If f(x) is a function with period P, then f(ax), where ‘a’ is non – zero real number, is periodic with period P/|a|.
Example: Find the periodic function f(x) = sin(6x)
Solution: Given f(x) = sin(6x)
If f(x) is a function with period P, then f(ax), where ‘a’ is non – zero real number, is periodic with period P/|a| = 2π/|a|
sin(6x) periodic function with period 2π/6 (since a = 6)
2. c f(x) is periodic with period T.
a) f (x + c) is periodic with period T.
b) f(x) + c is periodic with period T.
c) f(x) – c is periodic with period T.
d) If a constant is added, subtracted, multiplied, or divided in the periodic function, period remains the same.
e) Every constant function is always periodic, with no fundamental period.
f) Inverse of a periodic function does not exist. But in case of a trigonometric function since domain and range are restricted and defined, hence inverse exists.
3. If f (x) is periodic with period T, then
a) kf (cx + d) has period T/mod(c), hence period is affected by coefficient of x only.
b) If f(x) and g(x) are two functions with period T₁ and T₂ respectively, and h(x) = af(x) + b g(x), then h (x) has period = LCM of (T₁, T₂).
Example: Find the period of f(x) = sin (2πx + π/4) – 2sin (3 πx + π/3).
Solution: Period of f(x) = sin (2πx + π/4) – 2sin (3 πx + π/3)
Period of sin (2πx + π/4)
= 2π/2π (since a = 2π)
Period of 2sin (3 πx + π/3)
= 2π/3π (since a = 3 π)
Period of f(x) = L.C.M of 1 and ⅔
(L.C.M of numerator)/ H.C.F of Denominator = L.C.M of 1 and 2/H.C.F of 1 and 3 = 2/1 = 2
If f(x) is a periodic function with period T and g(x) is any function such that range of ‘f’ is a proper subset of the domain of g, then g(f(x)) is periodic with period T.