<< Chapter < Page Chapter >> Page >

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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
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
ya I also want to know the raman spectra
Bhagvanji
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

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play




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