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Drawings of graphs resulting from transformation applied by greatest integer function (GIF) follow the same reasoning and steps as deliberated for modulus operator. Before, we proceed to draw graphs for different function forms, we need to recapitulate the graph of greatest integer function (GIF) and also infer thereupon few of the values of GIF around zero.

Graph of gif

The GIF returns integral values.

For clarity, we apply circular symbols : solid circle to denote inclusion of point and empty circle to denote exclusion of point. Using solid circle is optional, but helps to identify points on the graph. The values of GIF around zero are (we can write these expressions by observing graph. A bit of practice to write down these intervals helps.) :

[ x ] = - 3 ; - 3 x < - 2 [ x ] = - 2 ; - 2 x < - 1 [ x ] = - 1 ; - 1 x < 0 [ x ] = 0 ; 0 x < 1 [ x ] = 1 ; 1 x < 2 [ x ] = 2 ; 2 x < 3 [ x ] = 3 ; 3 x < 4

Important point to note is that lower integer in the interval is included and higher integer is excluded. For negative interval like - 3 x < - 2 , note that -3 is lower end whereas -2 is higher end. Yet another feature of this function is that domain of the function is continuous in R. It means, there need to be a function value corresponding to all x.

A function like y=f(x) has different elements. We can apply GIF to these elements of the function. There are following different possibilities :

1 : y= f([x])

2 : y=[f(x)]

3 : [y]=f(x)

Greatest integer operator applied to independent variable

The form of transformation is depicted as :

y = f x y = f [ x ]

The graph of y=f(x) is transformed in y=f([x]) by virtue of changes in the argument values due to operation on independent variable. The independent variable of the function is subjected to greatest function operator. This changes the normal real value input to the function. Instead of real numbers, independent variable to function is rendered to be integers – depending on the value of x and interval it belongs to. A value like x= - 2.3 is passed to the function as -3 in the interval - 3 x < - 2 .

Clearly, real values of “x” are truncated to integer values in the interval of unity i.e. [-1,0), [0,1), [1.2) etc. It means that values of the function y=f(|x|) will remain same as that of its value corresponding to integral value of “x” till value of “x” changes to next interval. We need to apply modification to the curve to reflect this effect. Knowing that truncation takes place for successive integral values of x, we divide graph of y=f(x) to correspond to 1 unit segments of x-axis. For this, we draw lines parallel to y-axis at integral points along x-axis. From intersection point of lines drawn and function graph, we draw lines parallel to x-axis for the whole interval which extends for a unit value. This ensures that function values remain same to that of function value for the lower integral value of x in a particular interval of one.

From the point of construction of the graph of y=f([x]), we need to modify the graph of y=f(x) as :

1 : Draw lines parallel to y-axis (vertical lines) at integral values along x-axis to cover the graph of y=f(x).

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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
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

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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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