Periodic functions. Study of periodic functions

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Periodic functions are those in which the function values ​​(f (x) = y) repeat for certain values. of the variable x, that is, for each period determined by the values ​​of x, we will obtain repeated values ​​for the occupation.

Let's look at an example to better understand this definition:

Let's make a table with some values ​​for the variable x, listing the value of the function for each value of x.

x 0 1 2 3 4 5
f(x) 1 -1 1 -1 1 -1

Note that f (x)= 1 occurs only when the value of the variable x it's pair.
Note that f (x)= –1 occurs only when the value of the variable x is odd.

That is, this is a periodic function, in which we have two different periods, one in which the value of the function is 1 (f (x)= 1) and the other in which the function is –1 (f (x)= –1 ).

Note also that when x varies by two units, the value of the function is repeated, that is: f (x) = f (x+2) = f (x+4) = f (x+6)... Thus, we can say that the period of this function is 2.

Therefore, we can define periodic functions as follows:

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“A function is called periodic if there is a real number p > 0, such that: f (x)=f (x+p). Thus, the smallest value of p, which satisfies this equality, is called time course of the f” function.

Thus, if: f (x) = f (x+1.5) = f (x+3) = f (x+4.5), it is a periodic function whose period p = 1.5 .

In trigonometric functions, we have examples of periodic functions such as sine function, cosine function, tangent function.

Example:

y = cos x

See that the value 1 repeats in a period p = , and that the value y = 0 repeats in a period p = π.


By Gabriel Alessandro de Oliveira
Graduated in Mathematics
Brazil School Team

Source: Brazil School - https://brasilescola.uol.com.br/matematica/funcoes-periodicas.htm

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