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Square of an even or odd function

The square of even or odd function is always an even function.

Properties of derivatives

1: If f(x) is an even differentiable function on R, then f’(x) is an odd function. In other words, if f(x) is an even function, then its first derivative with respect to "x" is an odd function.

2: If f(x) is an odd differentiable function on R, then f’(x) is an even function. In other words, if f(x) is an odd function, then its first derivative with respect to "x" is an even function.

Composition of a function

Every real function can be considered to be composed from addition of an even and an odd function. This composition is unique for every real function. We follow an algorithm to prove this as :

Let f(x) be a real function for x R. Then,

f x = f x + f - x } - f - x }

Rearranging,

f x = 1 2 { f x + f - x } + 1 2 { f x f - x } = g x + h x

Now, we seek to determine the nature of functions “g(x)” and “h(x). For “g(x)”, we have :

g - x = 1 2 [ f x + f { - x } ] = 1 2 { f x + f x } = g x

Thus, “g(x)” is an even function.

Similarly,

h - x = 1 2 [ f x f { - x } ] = 1 2 { f x f x } = h x

Clearly, “h(x)” is an odd function. We, therefore, conclude that all real functions can be expressed as addition of even and odd functions.

Even and odd extensions of function

A function has three components – definition(rule), domain and range. What could be the meaning of extension of function? As a matter of fact, we can not extend these components. The concept of extending of function is actually not a general concept, but limited with respect to certain property of a function. Here, we shall consider few even and odd extensions. Idea is to complete a function defined in one half of its representation (x>=0) with other half such that resulting function is either even or odd function.

Even function

Let f(x) is defined in [0,a]. Then, even extension is defined as :

|f(x); 0≤x≤a g(x) = || f(-x); -a≤x<0

The graphical interpretation of such extension is that graph of function f(x) is extended in other half which is mirror image of f(x) in y-axis i.e. image across y-axis.

Odd extension

Let f(x) is defined in [0,a]. Then, odd extension is defined as :

| f(x); 0≤x≤a g(x) = || -f(x); -a≤x<0

The graphical interpretation of such extension is that graph of function f(x) is extended in other half which is mirror image of f(x) in x-axis i.e. image across x-axis.

Exercises

Determine whether f(x) is odd or even, when :

f x = e x + e - x

The function “f(x)” consists of exponential terms. Here,

f - x = e - x + e - - x = e - x + e x = e x + e - x = f x

Hence, given function is even function.

Determine whether f(x) is odd or even, when :

f x = x e x 1 + x 2

The function “f(x)” consists of exponential terms. In order to check polarity, we determine f(-x) :

f x = x e x 1 + x 2 = x 1 / e x 1 x 2

f x = x e x 1 e x x 2

We observe here that it might be tedious to reduce the expression to either “f(x)” or “-f(x)”. However, if we evaluate f(x) – f(-x), then the resulting expression can be easily reduced to simpler form.

f x f x = x e x 1 + x 2 + x e x 1 e x + x 2

f x f x = x e x 1 x e x e x 1 + x = x 1 e x e x 1 + x = 0

Hence,

f x = f - x

It means that given function is an even function.

) How to check whether a pulse equation of the form

y = a { 3 x + 4 t 2 + b }

is symmetric or asymmetric, here "a" and "b" are constants.

Posted by Dr. R.K.Singhal through e-mail

The pulse function has two independent variables “x” and “t”. The function needs to be even for being symmetric about y-axis at a given instant, say t =0.

We check the nature of function at t = 0.

y = a 9 x 2 + b

f - x = a { 9 - x 2 + b } = a 9 x 2 + b = f x

Thus, we conclude that given pulse function is symmetric.

Determine whether f(x) is odd or even, when :

f x = x 2 cos x | sin x |

The “f(x)” function consists of trigonometric and modulus functions. Here,

f - x = - x 2 cos - x | sin - x |

We know that :

- x 2 = x 2 ; cos - x = cos x ; | sin - x | = | - sin x | = | sin x |

Putting these values in the expression of f(-x), we have :

f - x = - x 2 cos - x | sin - x | = x 2 cos x | sin x | = f x

Hence, given function is an even function.

Determine whether f(x) is odd or even, when :

f x = x e - x 2 tan 2 x

The “f(x)” function consists of exponential terms having trigonometric function in the exponent. Here,

f - x = - x e - { - x 2 tan 2 - x }

We know that :

- x 2 = x 2 ; tan 2 - x = - tan x 2 = tan 2 x

f - x = - x e - { - x 2 tan 2 - x } = x e - x 2 tan 2 x = f x

Hence, given function is an odd function.

Questions & Answers

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
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
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
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
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.
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 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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