<< Chapter < Page Chapter >> Page >

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

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
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
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor 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