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Greatest Integer 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 [] Greatest Integer Function

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 $

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

Graph of Greatest Integer Function

Fractional Part of x

We have seen that
[1.56] =1
[.25]=0
[-5.45]=-5
[-.15]=-1
[-5]=-5
[1]=1
This can be written as
$ [x] \geq 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 {x} < 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
graph of fractional part function

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. $[x] \geq k \Rightarrow x \geq n , x \in [n,\infty)$
b. $[x] > k \Rightarrow x \geq n +1 , x \in [n+1,\infty)$
c. $[x] \leq k \Rightarrow x \leq n+1 , x \in (-\infty,n+1)$
d. $[x] < k \Rightarrow x < n , x \in (-\infty,n)$
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. $[x+y]=\begin{cases} [x] + [y] & \text{ if } \left\{x \right\} + \left\{y \right\} < 1 \\ [x]+[y]+1 & \text{ if } \left\{x \right\} + \left\{y \right\} \geq 1 \end{cases}$

Solved examples of Greatest Integer Function and Fractional part of x

1. Find the domain of the below function?
a. $y =\sqrt {[x] -1|}$
b. $y = \frac {1}{\sqrt { 2- [x]}}$
Solution

a. $y =\sqrt {[x] -1|}$
Clearly This is defined for $[x] -1 \geq 0$
or
$[x] \geq 1$
$ x \in [1,\infty)$
So Domain is $[1,\infty)$
b. $y = \frac {1}{\sqrt { 2- [x]}}$
Clearly This is defined for $2- [x] > 0$
or
2 > [x]
or [x] < 2
$ x \in (-\infty,2)$
So Domain is $(-\infty,2)$

2. Find the domain and range of function
$y= \frac {1}{\sqrt {x- [x]}}$
Solution
We have ,
$y= \frac {1}{\sqrt {x- [x]}}$
Domain of y
We know that
$0 \leq x -[x] < 1 , x \in R$
for $x \in Z ,x -[x]=0$
So Domain is R - Z
Range of y
$0 < x- [x] < 1 , x \in R -Z$
$ 0 < \sqrt {x -[x]} < 1$
$ 1 < \frac {1}{\sqrt {x- [x]}} < \infty$
So Range is = $(1,\infty)$

3. Let $f : R \rightarrow R$ , $f(x) = x^2 + 2[x] -1$ for each $x \in R$
Find the values of f(x) at x= 1.2 ,-.5 ,-2.1
Solution
$f(x) = x^2 + 2[x] -1$
$f(1.2) = (1.2)^2 + 2[1.2] -1 =1.44 +2-1=2.44$
$f(-.5) = (-.5)^2 + 2[-.5] -1 =.25 -2-1=-2.75$
$f(-2.1) = (-2.1)^2 + 2[-2.1] -1 =4.41 -6-1=-2.59$

Quiz Time

Question 1Find the domain of the function $f(x)= \frac {1}{\sqrt {[x]-x}}$
A. R - Z
B. R
C. {}
D. Z
Question 2Find the range of values of x for which $[x]^2 -4 =0$ $
A. $[-2,-1) \cup [2,3)$
B. $[-2,-1] \cup [2,3]$
C. $[-2,-3) \cup [2,3)$
D. $[-2,-3] \cup [2,3)$
Question 3 if $f(x) =[x]^2 +[x] +2$,then which of them is incorrect
A. f(1)=4
B. f(.75)=2
C. f(-.50)=2
D. f(0) =3
Question 4 Value of the expression [{x}] + {[y]} is
A. 1
B. 0
C. 2
D. Cannot be determined
Question 5 The value {-.75} where {.} is the fractional part is
A. -.75
B. .25
C. 0
D. None of these
Question 6The domain of the function f given by $f (x) = \frac {1}{\sqrt {x^2 -[x]^2}}$ is
A. Domain = R
B. Domain = R+
C. Domain = R+ -Z
D. Domain = Z



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