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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

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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
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
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?
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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Loga
what does nano mean?
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nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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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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