Q 133 - Precalculus Enhanced with Graphing Utilities

Question

Describe your understanding of the meaning of a periodic function. 

Step-by-Step Solution

Answer

A function $f\left(x\right)$ is said to be periodic, if there exists a positive real number $T$ such that

$f(x+T)=f\left(x\right)$.

The smallest value of $T$ is called the period of the function.

In this way we can say that a function is said to be periodic if it starts repeating its values after the any smallest period.

That means the graph of the function repeats its shape after any fixed period, then the function is said to be a periodic function.

Step 1. Given Information

We have to describe the meaning of periodic function.

Firstly we will give definition of periodic function and then by using mathematical notation of periodic function, we will explain it.

Step 2. Explanation of Periodic Function

A function is said to be periodic if  repeats its values at regular intervals A function $f\left(x\right)$ is said to be periodic, if there exists a positive real number $T$ such that

$f(x+T)=f\left(x\right)$.

The smallest value of $T$ is called the period of the function.

In this way we can say that a function is said to be periodic if it starts repeating its values after the any smallest period.

That means the graph of the function repeats its shape after any fixed period, then the function is said to be a periodic function.

The trigonometry function; sine function, cosine function, secant function and cosecant function are periodic functions of period $2\pi$.

Similarly tangent function and cotangent functions are also periodic functions of period $\pi$.