# Modulus Function

## Greatest Integer Function

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

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 Modulus 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 Modulus Function
The graph of the modulus 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 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]
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
$x-1 < [x] \leq x$
$-x \leq -[x] < 1-x$
$0 \leq x -[x] < 1$
$0 \leq {x} < 1$

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

## Properties of the Greatest Integer Function and Fractional part of x

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