Periodic Function

What periodic functions are

A function $ y=f(x) $ is said to be periodic with period $ T $, where $ T>0 $, if the condition $$ f(x+kT)=f(x) $$ is satisfied for every $ x $ in the domain and for every integer $ k $.

In a periodic function, the graph repeats itself at regular intervals equal to the period.

Consequently, if a function is periodic with period T, then it is also periodic with any period that is an integer multiple of T.

$$ f(x+T) = f(x+kT) = f(x) $$

This occurs because the values $ x $, $ x+T $, and $ x+kT $ are mapped to the same function value.

For example, if a function is periodic with period T, then it is also periodic with period 2T, 3T, 4T, and so forth. The smallest positive period T is called the fundamental period.

A practical example

The cosine function is a periodic function with period T=2π.

$$ y = \cos(x) $$

For instance, when x=0 and x=2π, the function f(x) takes the same value.

The same value is obtained again for x=4π. The same behavior continues for x=6π, and so on.

$$ \cos(0) = \cos(2 \pi ) = \cos (4 \pi ) $$

More generally, the function $ f= \cos (x) $ reproduces the same values in its range for every integer multiple of the period T.

$$ \cos (x) = \cos(x+kT) $$

Therefore, the cosine function is periodic with $ T= 2 \pi $.

Notes

Some additional remarks about periodic functions:

• A periodic function is not injective
  For a periodic function, there exists a period \( T \neq 0 \) such that \( f(x+T)=f(x) \) for every \( x \) in the domain. As a result, the same value in the codomain is assumed by infinitely many distinct elements of the domain, separated by integer multiples of \( T \). For this reason, a periodic function cannot be injective. Conversely, if a function is injective, then it cannot be periodic.

And so on.