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Graph of {x}

Few function expressions in different intervals are :

For - 2 x < - 1, f x = { x } = x [ x ] = x 2 = x + 2

For - 1 x < 0, f x = { x } = x [ x ] = x - 1 = x + 1

For 0 x < 1, f x = { x } = x [ x ] = x 0 = x

For 1 x < 2, f x = { x } = x [ x ] = x 1

For 2 x < 3, f x = { x } = x - [ x ] = x - 2

The graph of the function is shown here :

Graph of {x} function

The domain of the function is R.

We see that there is no restriction on values of x and as such its domain has the interval equal to that of real numbers. The fractional part function can only evaluate to non-negative values between 0≤y<1. Hence,

Domain = R

Range = 0 y < 1

FPF is a periodic function. The values are repeated with a period of 1. Further, function is defined for all real x, but graph is not continous. It breaks at integral values of x.

Least integer function

We have seen that greatest integer function represents the integer, which can be considered to be the floor integral value of a real number. Correspondingly, we define a ceiling function called “least integer function (LIF)”, which returns the least integer greater than or equal to the number (x). We denote least integer function as “[x)” or "(x)". Some authors reserve "(x)" for near integer function. It is not important as we can always specify what we mean by qualifying the symbol explicitly. We interpret LIF as :

  • [x) = least integer greater than or equal to the number x
  • [x) = least integer not less than or equal to the number x

Clearly, least integer function returns a value, which is the integral “ceiling” of the number. For this reason, least integer function is also known as “ceiling” function. Working rules for finding least integer function are :

  • If “x” is an integer, then [x) = x.
  • If “x” is not an integer, then [x) evaluates to least integer greater than “x”.

The value of f(x) is an integer (n) such that :

f x = n ; if n - 1 < x n n Z

Graph of least integer function

Few initial values of the functions are :

F o r - 3 < x - 2, f x = [ x ) = - 2

F o r - 2 < x - 1, f x = [ x ) = - 1

F o r 1 < x 0, f x = [ x ) = 0

F o r 0 < x 1, f x = [ x ) = 1

F o r 1 < x 2, f x = [ x ] = 2

Graph of least integer function

The domain of the function is R.

We see that there is no restriction on values of “x” and as such its domain has the interval equal to that of real numbers. On the other hand, the least integer function evaluates only to integer values. It means that the range of the function is set of integers, denoted by "Z". Hence,

Domain = R

Range = Z

GIF is not a periodic function. Though function is defined for all real x, but graph is not continous. It breaks at integral values of x.

Important properties

Certain properties of least integer function are presented here :

1: If and only if “x”is an integer, then :

[ x ) = x

2: If and only if at least either “x” or “y” is an integer, then :

[ x + y ) = [ x ) + [ y )

For example, let x = 2.27 and y = 0.63. Then,

[ x + y ) = [ 2.27 + 0.63 ) = [ 2.9 ) = 3

[ x ) + [ y ) = [ 2.27 ) + [ 0.63 ) = 3 + 1 = 4

However, if one of two numbers is integer like x = 2 and y = 0.63, then the proposed identity as above is true.

4: If “x” belongs to integer set, then :

[ x ) + [ - x ) = 0 ; x Z

For example, let x = 2.Then

[ 2 ) + [ - 2 ) = 2 2 = 0

We can use this identity to test whether “x” is an integer or not?

3: If “x” does not belong to integer set, then :

[ x ) + [ - x ) = + 1 ; x Z

For example, let x = 2.7.Then

[ 2.7 ) + [ - 2.7 ) = 3 2 = + 1

Nearest integer function

Nearest integer function, as the name suggests, returns the nearest integer. It is denoted by the symbol, "(x)".

The value of "(x)" is an integer "n" such that :

f x = ( x ) = n ; if n x n + 1 / 2, n Z

f x = n + 1 ; if n + 1 / 2 < x n + 1, n Z

Examples :

2.3 = 2, 2.6 = 3

- 2.3 = - 2, - 2.6 = - 3


Find domain of the function :

f x = x 2 [ x ] 2

We analyze given function using its properties to find domain. Subsequently, we shall use graphical solution, which is more elegant. Now, for radical function,

x 2 [ x ] 2 0

Evaluation of this expression for integer values of x is easy. We know that [x] evaluates to x for all integer values of x :

[ x ] = x ; x Z

Squaring both sides,

[ x ] 2 = x 2 ; x Z x 2 [ x ] 2 = 0 ; x Z

However, evaluation of expression is slightly difficult for other values of x. Now, consider positive interval 1≤x<2. Here, [x] evaluates to 1 and its square is 1, which is less than or equal to x 2 . On the other hand, in negative interval -2≤x<-1, [x] evaluates to -2 and its square is 4, which is equal to or greater than x 2 .

[ x ] 2 x 2 ; x > 0 [ x ] 2 x 2 ; x > 0

Note that we have included “equal to sign” for both intervals of x. Equal to sign is appropriate when x is integer. For x=0, expression evaluates to 0. It means expression is non-negative for all non-negative x. But expression also evaluates to 0 for negative integers. Hence, domain of given function is :

Domain = 0, { - n ; n N }

Graphical analysis

We draw y = [ x ] and y = [ x ] 2 as in the first and second figures. Finally, we superimpose y = x 2 on the graph y = [ x ] 2 as shown in the third figure. Noting values of x for which value of x 2 is greater than or equal to [ x ] 2 , the domain of the function is :


Domain is chosen for x such that difference of graphs is non-negative.

Domain = 0, { - n ; n N }

Questions & Answers

how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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s. Reply
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
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abeetha Reply
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Himanshu Reply
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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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silver nanoparticles could handle the job?
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I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
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how did you get the value of 2000N.What calculations are needed to arrive at it
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Period of sin^6 3x+ cos^6 3x
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Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
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