# Mathematics: Relations and Functions-I: Some Special Functions

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## Some Special Functions

**Monotonic Function:**

Let be a function then is said to be monotonic on an interval

if it is either increasing or decreasing on that interval.

For function to be increasing on an interval

and for function to be decreasing on

A function may not be monotonic on the whole domain, but it can be on different intervals of the domain.

Example:

Consider the function defined by .

Solution:

Now

is a __Monotonic Function__ on .

It is only increasing function on this interval

But

is a __Monotonic Function__ on.

It is only decreasing function on this interval

**Even Function:**

A function is said to be an even function if for each of domain.

Example:

(1)

(2)

(3)

Solution:

(1) If then.

(2) If then.

(3) If then.

The graph of this even function (modulus function) is shown in the figure above.

**Odd Function:**

A function is said to be an odd function if for each

Example:

(1)

(2)

Solution:

(1) If then.

(2) If then.

**Greatest Integer Function (Step Function):**

which is the greatest integer less than or equal to . is called Greatest Integer Function or Step Function.

Example:

Let us draw the graph of

Solution:

Domain of the step function is the set of real numbers.

Range of the step function is the set of integers.