# 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 = $$2\pi$$