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

Fraction part function

FPF is a periodic function with period 1.

The values of FPF around zero are (we can write these expressions by observing graph. A bit of practice to write down these intervals helps.) :

{ x } = x + 2 ; - 2 x < - 1 { x } = x + 1 ; - 1 x < 0 { x } = x ; 0 x < 1 { x } = x 1 ; 1 x < 2

Important point to note is that lower integer is included, but higher integer is excluded in the intervals of unity in which FPF is defined. The graph segment in the interval [0,1) is y=x i.e. identity function. We obtain expression of function in right intervals (positive value intervals of x) by shifting identity function towards right by 1 successively and in left intervals (negative value intervals of x) by shifting identity function towards left by 1 successively. The values of FPF are continuous real values which is equal to or greater than zero but less than 1. These function values are repeated in each of intervals of unity along x-axis. Thus, FPF is a periodic function with period of 1. Domain of FPF is R and range is [0,1). Further, FPF is related to real number as x=[x]+{x}.

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

1 : y = f({x})

2 : y ={f(x)}

3 : {y} = f(x)

Fraction part operator applied to the argument

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. The independent variable is subjected to fraction part operator. This changes the normal real value input to function. Instead of real numbers, independent variable to function is rendered to be fractions irrespective of values of x. A value like x = - 2.3 is passed to the function as 0.7 in the interval [0,1).

Clearly, real values of “x” are truncated to fraction values in all intervals. It means that same set of values of the function y=f(|x|) corresponding to interval of x defined by [0,1] will repeat in other intervals along x-axis. The FPF is a periodic function with a period of 1. Taking advantage of this fact, we obtain graph of y=f({x}) by repeating part of graph for x in [0,1) to other intervals along x-axis. Clearly, transformed function y=f({x}) is periodic with a period of 1.

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).

2 : Identify part of the graph for values of x in [0,1). Include end point corresponding to x=0 and exclude end point corresponding to x=1.

3 : Repeat the part of the graph identified in step 2 for other intervals of x

Questions & Answers

what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
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Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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Renato
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Stoney Reply
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biomolecules are e building blocks of every organics and inorganic materials.
Joe
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research.net
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sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
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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.
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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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.
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for screen printed electrodes ?
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of graphene you mean?
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in general
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Graphene has a hexagonal structure
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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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