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Identity function

The dependent (y) and independent (x) variables have same value. Identity function is similar in concept to that of identity relation which consists of relation of an element of a set with itself. It is a linear function in which m=1 and c=0. Identity function form is represented as :

y = f(x) = x

Identity function

Identity function is a polynomial of degree 1.

The graph of identity function is a straight line bisecting first and third quadrants of coordinate system. Note that slope of straight line is 45°. It is clear from the graph that its domain and range both are real number set R.

Quadratic function

The general form of quadratic function is :

f(x) = a x 2 + b x + c ; a , b , c R ; a 0

We shall discuss quadratic function in detail in a separate module and hence discussion of this function is not taken up here.

Graph of polynomial function

Graph of polynomial is continuous and non-periodic. If degree is greater than 1, then it is a non-linear graph. Polynomial graphs are analyzed with the help of function properties like intercepts, slopes, concavity, and end behaviors. The may or may not intersect x-axis. This means that it may or may not have real roots. As maximum number of roots of a polynomial is at the most equal to the order of polynomial, we can deduce that graph can at the most intersect x-axis “n” times as maximum numbers of real roots are “n”.

The fact that graph of polynomial is continuous suggests two interesting inferences :

1: If there are two values of polynomial f(a) and f(b) such that f(a)f(b)<0, then there are at least 1 or an odd numbers of real roots between a and b. The condition f(a)f(b)<0 means that function values f(a) and f(b) lie on the opposite sides of x-axis. Since graph is continuous, it is bound to cross x-axis at least once or odd times. As such, there are at least 1 or odd numbers of real roots (as shown in the left figure down).

Roots of polynomial function

The numbers of x-intercepts depend on nature of product given by f(a)f(b).

2 : If there are two values of polynomial f(a) and f(b) such that f(a)f(b)>0, then there are either no real roots or there are even numbers of real roots between a and b. The condition f(a)f(b)>0 means that function values f(a) and f(b) are either both negative or both positive i.e. they lie on the same side of x - axis. Since graph is continuous, it may not cross at all or may cross x-axis even times (as shown in the right figure above). Clearly, there is either no real root or there are even numbers of real roots.

We shall study graphs of quadratic polynomials in a separate module. Further, other graphs will be discussed in appropriate context, while discussing a particular function. Here, we present two monomial quadratic graphs y = x 2 and y = x 3 . These graphs are important from the point of view of generalizing graphs of these particular polynomial structure. The nature of graphs y = x n , where “n” is even integer greater than equal to 2, is similar to the graph of y = x 2 . We should emphasize that the shape of curve simply generalizes the nature of graph – we need to draw them actually, if we want to draw graph of a particular monomial function. However, we shall find that these generalizations about nature of curve lets us know a great deal about the monomial polynomial. In particular, we can conclude that their domain and range are real number set R.

Even degree function

The nature of graphs of degree of positive even integer are similar to the graph shown.

Similarly, the nature of graphs y = x n , where “n” is odd number integer greater than 2, is similar to the graph of y = x 3 .

Odd degree function

The nature of graphs of degree of positive odd integer are similar to the graph of shown.

Questions & Answers

how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
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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 ?
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.
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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.
Do you know which machine is used to that process?
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for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
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Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Period of sin^6 3x+ cos^6 3x
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Period of sin^6 3x+ cos^6 3x
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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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