Periodic function:

A function f: X→Y is called a periodic function given there exists a positive real number T such that f(x + T) = f(x), for all x ∈ X. The least of all such positive numbers T is known as the principal period or basic period or simply period of f.

• To examine the periodicity of a function put f(T+x)=f(x) and compute that equation to calculate the positive values of t independent of x. If positive numbers of T independent of x are calculated, then f(x) is a periodic function and the least positive value of T is the period of the function f(x). If no positive number value of T independent of x is calculated then f(x) is non-periodic function.

• A constant function is typically periodic but would not have a well-defined period.

• If f(x) is basic periodic with period p, then f(ax + b) where  a, b ∈ R (a ≠ 0) is also period with period p/|a|.

• If f(x) is basic periodic with period p, then a f(x) + b where a, b ∈ R (a ≠ 0) is also periodic with period p.

• If f(x) is periodic with period p, then f (ax + b) where a, b ∈ R (a ¹ 0) is also period with period p/|a|

• Suppose f(x) has period p = m/n (m, n ∈ N and co-prime) and g(x) has period q = r/s (r, s ∈ N and co-prime) and take t be the LCM of p and q i.e. t = LCM of (m,r)/HCM of (r,s), then t will be the period of f + g given there does not exist a positive number) k (< t) for which f(k + x) + g(k + x) = f(x)+ g(x), else k will be the period. The same principle is c applicable for any other algebraic combination of g(x) and f(x).

SOME IMPORTANT POINTS:

• LCM of p and q forever exist if p/q is a rational quantity. If p/q is irrational then algebraic) combination of g and f is non-periodic.

• sinⁿx, cosⁿx, cosecⁿx and Secⁿx have period 2Π if n is odd and Π if n is even.

• tanⁿx and cotⁿx have period Π whether n is odd or even.

• If g is periodic then fog may always be a periodic function. Period of fog can or cannot be the period of g.

• If f is periodic and g is exactingly monotonic another than linear then fog is non-periodic.

Problem: Calculate the period of function sin4x + tan2x.

Solution: Period of sin4x is Π/2, also period of tan 2x is Π/2.

Therefore period of f(x) is Π/2

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