What is a Periodic Function – Definition and Example

Here, you will learn what is a periodic function with definition and example.

Let’s begin –

Periodic Function

A function f(x) is called periodic if there exist a positive number T (T > 0), where T is the smallest such value called the period of the function such that f(x + T) = f(x), for all values of x, x + T within the domain of f.

Note :

(i)  Odd powers of sinx, cosx, secx, cosecx are periodic with period 2\(\pi\).

(ii)  None zero integral powers of tanx, cotx are periodic with period \(\pi\).

(iii)  Non zero even powers or modulus of sinx, cosx, secx, cosecx are periodic \(\pi\).

(iv)  f(T) = f(0) = f(-T), where ‘T’ is the period.

(v)  if f(x) has period T then f(ax + b) has a period T/|a| (a \(\ne\) 0).

(vi)  If f(x) & g(x) are periodic with period \(T_1\) & \(T_2\) respectively, then period of f(x) \(\pm\) g(x) is L.C.M of (\(T_1\), \(T_2\))

(vii)  Every constant function is always periodic.

(viii)  Inverse of a periodic functions does not exist.

Example : Find the periods of the function f(x) = \(e^{ln(sinx)}\) + \(tan^3x\) – cosec(3x – 5)

Solution : Period of \(e^{ln(sinx)}\) = \(2\pi\), \(tan^3x\) = \(\pi\)

cosec(3x – 5) = \(2\pi\over 3\)

\(\therefore\)   Period = \(2\pi\)

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