A function is said to be periodic if there exists a positive real number “T” such that

where “D” is the domain of the function f(x). The least positive real number “T” (T>0) is known as the fundamental period or simply the period of the function. The “T” is not a unique positive number. All integral multiple of “T” within the domain of the function is also the period of the function. Hence,

In the context of periodic function, an “aperiodic” function is one, which in not periodic. On the other hand, a function is said to be anti-periodic if :

**Periodicity and period **

In order to determine periodicity and period of a function, we can follow the algorithm as :

- Put f(x+T) = f(x).
- If there exists a positive number “T” satisfying equation in “1” and it is independent of “x”, then f(x) is periodic. Otherwise, function, “f(x)” is aperiodic.
- The least value of “T” is the period of the periodic function.

#### Example 1

Problem : Let f(x) be a function and “k” be a positive real number such that :

Prove that f(x) is periodic. Also determine its period.

Solution : The given equation can be re-written as :

Here, our objective is to convert RHS of the equation as f(x). For this, we need to substitute "x" such that RHS function acquires RHS function form. Replacing “x” by “x+k”, we have :

Combining two equations,

It means that f(x) is a periodic function and its period is “2k”.

#### Example 2

Problem : Determine period of the function :

Solution : The function is sum of two trigonometric functions. We can reduce this function is terms of a single trigonometric function to determine its periodic nature. Let

Substituting in the function, we have :

This is a periodic function. Also, period of “ag(x)” is same as that of “g(x)”. Therefore, period of “r sin (kx + θ)” is same as that of “sin (kx + θ)”. On the other hand, period of g(ax+b) is equal to the period of g(x), divided by “|a|”. Now, period of “sinx” is “2π”. Hence, period of the given function is :

Alternatively, we can treat given function as addition of two functions. The period of each term is “2π/|k|”. Applying LCM rule (discussed later), the period of given function is equal to LCM of two periods, which is “2π/|k|”.