Greatest Integer Function & Fractional Part Function

Greatest Integer Function Definition

Greatest Integer Function is defined as the real valued function $f : R \rightarrow R$ , y = [x] for each $x \in R$
For each value of x, f(x) assumes the value of the greatest integer, less than or equal to x.

It is also called the Floor function and step function
Symbol of Greatest Integer Function is []

Example
[2.56] =2
[.5]=0
[-6.45]=-7
[-.65]=-1

This can be generalized as
$[x] = –1 , –1 \leq x < 0$
$[x] = 0 , 0 \leq x < 1$
$[x] = 1 , 1 \leq x < 2$
$[x] = 2 , 2 \leq x < 3 $

Question

Solution

Domain and Range of the Greatest Integer Function

For $f : R \rightarrow R , y = f(x) = [x]$ for each $x \in R$

Domain = R
Range = Z (integer set) as it only attains integer values

Graph of the Greatest Integer Function

The graph of the Greatest Integer Function is shown below. It coincide with the graph of identity function y=x for $x \geq 0$ and for x< 0, it is a graph of linear function y=-x

Fractional Part Function(Fractional part of x)

We have seen that
[1.56] =1
[.25]=0
[-5.45]=-6
[-.15]=-1
[-5]=-5
[1]=1
This can be written as
$ [x] \leq x$

Fractional part of x is defined as the difference between [x] and x. It is denoted as {x}
So,
{x}=x -[x]
Symbol of Fractional part of x is {} Example
x=4.65 ,then [x]=4 ,{x}=.65
x= -.65, then [x]=-1 ,{x}= .35 ( This is very important thing to remember for negative number)
x =-1.45 ,then [x]=-2 ,{x}= .55 ( This is very important thing to remember for negative number)
x=3 ,then [x]=3 ,{x}=0

We know that
$ x-1 < [x] \leq x$
$ -x \leq -[x] < 1-x$
Adding x
$ 0 \leq x -[x] < 1$
$ 0 \leq \left \{ x \right \} < 1 $

So Fractional part {x} is always non-negative and lies between (0,1).

So we can define Fractional part function as
y=f(x) = {x} =x -[x]

This is good for $x \in R$

Range of the fractional part function is [0,1)
Graph of the fractional part function is given below

Question

Solution

Greatest Integer Function Properties and Fractional part of x Properties

1. If x is an integer

[x]=x
{x} =0

2. For $x \in R$ [[x]]= [x]
[{x}]=0
{[x]}=0

3. for $k \in Z$
a. $[x] \geq k \Rightarrow x \geq k , x \in [k,\infty)$
b. $[x] > k \Rightarrow x \geq k +1 , x \in [k+1,\infty)$
c. $[x] \leq k \Rightarrow x \leq k+1 , x \in (-\infty,k+1)$
d. $[x] < k \Rightarrow x < k , x \in (-\infty,k)$
e. $[x + k]=[x] +k$

4.
a. $y=[x]+ [-x]=\begin{cases} 0 & \text{ if } x \in Z \\ -1 & \text{ if } x \notin Z \end{cases}$
b. $y=[x]- [-x]=\begin{cases} 2[x]+1 & \text{ if } x \notin Z \\ 2[x]] & \text{ if } x \in Z \end{cases}$
c. $y=\left\{ x \right\}- \left\{-x\right\}=\begin{cases} 0 & \text{ if } x \in Z \\ 1 & \text{ if } x \notin Z \end{cases}$
3. For real number x and y
a. [x +y] = [x] + [y+x - [x] ]
b. $[x+y]=\begin{cases} [x] + [y] & \text{ if } \left\{x \right\} + \left\{y \right\} < 1 \\ [x]+[y]+1 & \text{ if } \left\{x \right\} + \left\{y \right\} \geq 1 \end{cases}$





Also Read

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• Notes
  • Cartesian Products
  • What is relations
  • What is Function
  • Domain of Function
  • Range of Function
  • Identity Function
  • Constant Function
  • Linear Function
  • Modules Function
  • Greatest Integer Function
  • Polynomial Function
  • Algebra of Real Function
• Assignments
  • Relations and Functions Class 11 Worksheet
  • NCERT Solutions Relation and Functions class 11 Exercise 2.1
  • NCERT Solutions Relation and Functions class 11 Exercise 2.2
  • NCERT Solutions Relation and Functions class 11 Exercise 2.3

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