Periodic Function
Periodic function is a function that repeats itself at regular intervals. The period of a function is an important characteristic of periodic functions, which helps to define a function. A periodic function y = f(x), having a period P, can be represented as f(X + P) = f(X).
Let us learn more about the periodic function, properties of periodic functions, and examples on periodic functions.
What Is a Periodic Function?
A function y= f(x) is said to be a periodic function if there exists a positive real number P such that f(x + P) = f(x), for all x belongs to real numbers. The least value of the positive real number P is called the fundamental period of a function. This fundamental period of a function is also called the period of the function, at which the function repeats itself.
f(x + P) = f(x)
The sine function is a periodic function with a period of 2π. Sin(2π + x) = Sinx.
The following are the graphs of some of the periodic functions. The graph of each of the below periodic functions has translational symmetry.
Periods of Some Important Periodic Functions
The period of a function helps us to know the interval, after which the range of the periodic function repeats itself. The domain of a periodic function f(x) includes of the real number values of x, the range of a periodic function is a limited set of values within an interval. The length of this repeating interval, or the interval after which the range of the function repeats itself, is called the period of the periodic function.
The period of some of the important periodic functions are as follows.
 The period of Sinx and Cosx is 2π.
 The period of Tanx and Cotx is π.
 The period of Secx and Cosecx is 2π.
Properties of Periodic Functions
The following properties are helpful for a deeper understanding of the concepts of a periodic function.
 The graph of a periodic function is symmetric and repeats itself along the horizontal axis.
 The domain of the periodic function includes all the real number values, and the range of the periodic function is defined for a fixed interval.
 The period of a periodic function, against which the period repeats itself is equal to the constant across the entire range of the function.
 If f(x) is a periodic function with a period of P, then 1/f(x) will also be a periodic function with the same fundamental period P.
 If f(x) is a periodic function with a period of P, then f(ax + b) is also a periodic function with a period of P/a.
 If f(x) is a periodic function with a period of P, then af(x) + b is also a periodic function with a period of P.
Some Important Periodic Functions
The following are some of the advanced periodic functions, which can be explored further.
Euler's Formula: The complex number formula e^{ix} = Coskx + iSinkx is made up of cosine and sine functions, which are periodic functions. Here these two functions are periodic, and the euler's formula represents a periodic function and has a period of 2π/k.
Jaccobi Elliptic Functions: The graph of these functions is elliptical in shape, rather than a circle, which is often seen for trigonometric functions. These elliptical shapes arise because of the involvement of two variables together, such as the amplitude and speed of a moving body, or the temperature and viscosity of the substance. These functions are commonly used in the description of the motion of a pendulum.
Fourier Series: The Fourier series is a superposition of various periodic wave function series to form a complex periodic function. It is usually composed of sine and cosine functions, and the summation of these wave functions is taken by assigning respective weight components to these series. The Fourier series has applications in the representation of heatwaves, vibration analysis, quantum mechanics, electrical engineering, signal processing, image processing.
Related Topics
The following topics help in a better understanding of periodic functions.
Examples on Periodic Function

Example 1:Find the period of the function Sin(4x + 5).
Solution:
The given function is Sin(4x + 5)
The period of Sinx is 2π., and the period of Sin(4x + 5) is 2π/4 = π/2.
Therefore, the period of Sin(4x + 5) is π/2.

Example 2: Find the period of Tan3x + Sin 5x/2
Solution:
The given function is f(x) = Tan3x + Sin5x/2
The period of Tanx is π, and the period of Tan3x is π/3.
The period of Sinx is 2π, and the period of Sin5x/2 is 2π/5/2 = 4π/5.
The calculation of the period of the function f(x) = tan3x + Sin5x/2, is as follows.
Period of f(x) = (LCM of π and 4π)/(HCF of 3 and 5) = 4π/1 = 4π.
Therefore, the period of Tan3x + Sin5x/2 is 4π.
FAQs on Periodic Function
What Is Periodic Function?
A function y= f(x) is said to be a periodic function if there exists a positive real number P such that f(x + P) = f(x), for all x belongs to real numbers. The least value of the positive real number P is called the fundamental period of a function. This fundamental period of a function is also called the period of the function, at which the function repeats itself. f(x + P) = f(x)
How Do You Know If a Function Is a Periodic Function?
A function can be identified as a periodic function if the range of the function repeats itself at regular intervals, and the function is of the form f(X + P) = F(X).
What Is a Periodic Function Formula?
The formula for a periodic function is f(X + P) = F(x). Here the function repeats itself, for different domains .
What Are the Examples of Periodic Function?
The examples of periodic function are the trigonometric functions, inverse trigonometric function, hyperbolic functions, and all the functions which represent periodic or circular motions in physics.
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