<< Chapter < Page Chapter >> Page >

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

Exercise

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

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

Domain = 0, { - n ; n N }

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
What is power set
Satyabrata Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Functions' conversation and receive update notifications?

Ask