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2 : Identify points of intersections of graph with parallel lines drawn in the earlier step.

3 : Draw lines of 1 unit parallel to x-axis from intersection points in the direction of positive x. The line ends at the next parallel line on right. Include intersection point but exclude other end of the line. Include transformation for all points of the graph.

The lines drawn in step 3 is the graph of y=f([x]).

Problem : Draw the graph of sin[x].

Solution : Following the construction steps, graph of y=sin[x]is drawn as shown here.

Graph of y=sin[x]

The argument of function is modified by GIF.

Problem : Draw graph of tan⁻¹[x], x∈[-2, 2].

Solution : Following the construction steps, graph of y= tan⁻¹ [x]is drawn as shown here.

Graph of y= tan⁻¹ [x]

The argument of function is modified by GIF.

See that function value corresponding to x=2 and x=-2 are not included in the preceding interval on the graph. As such, we need to put a solid circle at x=2 and x=-2 additionally. Further, we need to remove original graph of y= tan⁻¹ x (this step is not shown in the figure above).

Greatest integer operator applied to the function

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 applying changes to the output of the function. Whatever be the function values, they will be changed to integral values following definition of greatest integer values as given earlier for few intervals. Clearly, real values of “f(x)” are truncated to integer values in the interval of unity i.e. [-1,0), [0,1), [1.2) etc along y-axis.

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 x-axis (horizontal lines) at integral values along y-axis to cover the graph of y=f(x).

2 : Identify points of intersections of graph with parallel lines drawn in the earlier step. Draw lines parallel to y-axis (vertical lines) from the intersection points identified.

3 : Take x-projection of curve from the point of intersection between two consecutive vertical lines such that it lies on horizontal line of lower value. Include intersection point but exclude other end of the line. Further include points not covered by the projection.

The lines drawn in step 3 is the graph of y=[f(x)].

Problem : Draw the graph of [2sinx].

Solution : Following the construction steps, graph of y=[2sinx]is drawn as shown here.

Graph of y=[2sinx]

The value of function is modified by GIF.

Values assigned to greatest integer function

The form of transformation is depicted as :

y = f x [ y ] = f x

We need to evaluate this equation on the basis of assignment to the dependent expression. The value of function f(x) is first calculated for a given value of x. The value so evaluated is assigned to the GIF function [y]. We interpret assignment to [y]in accordance with the interpretation of equality of the GIF function to a value. In this case, we know that :

[ y ] = f x ; f x Z GIF can not be equated to non-integers. No solution.

[ y ] = f x ; f x Z y = Continuous interval of 1 unit starting from f(x)

Clearly, we need to neglect plot corresponding to all non-integral values of f(x). For every value of x, which yields integral value of f(x), there are multiple values of dependent expression [y] in an interval of 1 unit. For example, for [ y ] = f x = 2, y 2 y < 3 . In the nutshell, this graph is not continuous. There is no value of y corresponding to non integer f(x) and there are multiple values of y in an interval of 1 for integral values of f(x).

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 x-axis (horizontal lines) at integral values along y-axis to cover the graph of y=f(x).

2 : Identify points of intersections of graph with parallel lines (horizontal lines) drawn in the earlier step.

3 : Draw lines of 1 unit parallel to y-axis (vertical lines) from intersection points in the positive y-direction. Include intersection point but exclude other end of the line.

The lines drawn in step 3 is the graph of [y]= f(x).

Problem : Draw graph of [y]=(x+1)(x-2).

Solution : We first draw the graph of quadratic polynomial function y = x + 1 x 2 = x 2 x 2 . The lowest point of the parabola is calculated as :

D = - 1 2 4 X 1 X - 2 = 1 + 8 = 9

y min = - D 4 a = - 9 4 X 1 = - 2.25

Following construction steps, graph of [y]=(x+1)(x-2) is drawn as shown here.

Graph of [y]=(x+1)(x-2)

The value of expression is assigned to GIF.

Questions & Answers

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
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
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
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
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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

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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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