PERIODIC FUNCTIONS

Periodic Functions:

A function f(x) is said to be a periodic function if there exists a positive real number T such that f (x+ T) = f (x) for all x ϵ R.

We know that

Sin (x + 2x) = sin (x + 4x) = … = sin x

And,

cos (x + 2x) = cos (x + 4x) = … = cos x for all x ϵ R.

Therefore, sin x and cos x are periodic functions.

PERIOD:

If  f(x) is a periodic function, then the smallest positive real number T is called the period or fundamental period of function f(x) if f(x + T) = f(x) for all x ϵ R.

In order to check the periodicity of a function f(x), we follow the following algorithm.

ALGORITHM

STEP 1:

Write f(x + T) = f(x)

STEP 2:

Simplify the equation obtained in step 1 and solve it for T.

STEP 3:

If the values of T obtained in Step 2 are positive and independent of x, then f(x) is periodic, otherwise not.

STEP 4:

If f(x) is periodic, then choose the smallest value of T obtained in step 3. The value of T so obtained is the period of f(x)

Example:

Prove that the function f(x) = x – [x] is a periodic function. Also, find its period.

Solution:

Let T be a positive real number. If possible, let f(x) be periodic with period T. Then,

f (x + T) = f (x) for all x ϵ R

=> x + T – [x + T] = x – [x] for all x ϵ R

=> [x + T] – [x] = T for all x ϵ R

=> T = 1, 2, 3, 4,….. [·.· [x+n] – [x] = n for all n ϵ N.

Thus, there exists T > 0 such that f(x + T) = f(x) for all x ϵ R.

So, f (x) is a periodic function.

The smallest value of T satisfying f(x + T) = f (x) for all x ϵ R is 1.

Hence, f (x) = x – [x] has period 1.