Question

how to find periodicity of a function ?​

Answer 1

Step-by-step explanation:

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. ...

★The least value of “T” is the period of the periodic function.

Answer 2

Definition of periodic function

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

f(x+T)=f(x)for allx∈D

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,

f(x+nT)=f(x);n∈Z,for allx∈D

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 :

f(x+T)=−f(x)for allx∈D

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.

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

f(x+k)+f(x)=0for allx∈R

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

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

f(x+k)=−f(x)for allx∈R

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 :

⇒f(x+2k)=−f(x+k)for allx∈R

Combining two equations,

⇒f(x+2k)=−1X−f(x)=f(x)for allx∈R

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