# Periodic Functions

Function y=f(x) is called **periodic**, if exists such number `T!=0`, that for any x from domain of the function we have the following: `f(x+T)=f(x)=f(x-T)`.

Number T is called **period** of the function y=f(x).

From definition it follows that periodic functions has infinitely many periods. For example, if T is period of function, then any number of form kT, where k is integer number, is also period of function.

Often (but not always) among set of positive periods of the function we can find the smallest one. This period is called **main period**.

Usually, only main period is considered and it is called simply "period".

Examples of periodic functions (with specifying of main period):

- `y={x}` -period T=1; (see note function y={x})
- `y=sin(x)` -period `T=2pi`; (see note properties and graph of the function y=sin(x))
- `y=tan(x)` -period `T=pi`. (see note properties and graph of the function y=tan(x))

**Fact**. If function f is periodic and has period T, then function `Af(kx+b)` (where A,k and b are constant, `k!=0`) is also periodic with period `T/(|k|)`.

For example, period of the function `2sin(3x-pi/6)` is `(2pi)/3`; period of the function `tan(-x/3+pi/4)` is `pi/(|-1/3|)=3pi`.