# 5.1 Even and odd functions  (Page 3/3)

 Page 3 / 3

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 $\in$ R. Then,

$f\left(x\right)=f\left(x\right)+f\left(-x\right)\right\}-f\left(-x\right)\right\}$

Rearranging,

$f\left(x\right)=\frac{1}{2}\left\{f\left(x\right)+f\left(-x\right)\right\}+\frac{1}{2}\left\{f\left(x\right)-f\left(-x\right)\right\}=g\left(x\right)+h\left(x\right)$

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

$⇒g\left(-x\right)=\frac{1}{2}\left[f\left(-x\right)+f\left\{-\left(-x\right)\right\}\right]=\frac{1}{2}\left\{f\left(-x\right)+f\left(x\right)\right\}=g\left(x\right)$

Thus, “g(x)” is an even function.

Similarly,

$⇒h\left(-x\right)=\frac{1}{2}\left[f\left(-x\right)-f\left\{-\left(-x\right)\right\}\right]=\frac{1}{2}\left\{f\left(-x\right)-f\left(x\right)\right\}=-h\left(x\right)$

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 :

$f\left(x\right)={e}^{x}+{e}^{-x}$

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

$⇒f\left(-x\right)={e}^{-x}+{e}^{-\left(-x\right)}={e}^{-x}+{e}^{x}={e}^{x}+{e}^{-x}=f\left(x\right)$

Hence, given function is even function.

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

$f\left(x\right)=\frac{x}{{e}^{x}-1}+\frac{x}{2}$

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

$f\left(-x\right)=-\frac{x}{{e}^{-x}-1}+\frac{-x}{2}=-\frac{x}{1/{e}^{x}-1}-\frac{x}{2}$

$⇒f\left(-x\right)=-\frac{x{e}^{x}}{1-{e}^{x}}-\frac{x}{2}$

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.

$f\left(x\right)-f\left(-x\right)=\frac{x}{{e}^{x}-1}+\frac{x}{2}+\frac{x{e}^{x}}{1-{e}^{x}}+\frac{x}{2}$

$⇒f\left(x\right)-f\left(-x\right)=\frac{x}{{e}^{x}-1}-\frac{x{e}^{x}}{{e}^{x}-1}+x=\frac{x\left(1-{e}^{x}\right)}{{e}^{x}-1}+x=0$

Hence,

$f\left(x\right)=f\left(-x\right)$

It means that given function is an even function.

) How to check whether a pulse equation of the form

$y=\frac{a}{\left\{{\left(3x+4t\right)}^{2}+b\right\}}$

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.

$⇒y=\frac{a}{\left(9{x}^{2}+b\right)}$

$⇒f\left(-x\right)=\frac{a}{\left\{9{\left(-x\right)}^{2}+b\right\}}=\frac{a}{\left(9{x}^{2}+b\right)}=f\left(x\right)$

Thus, we conclude that given pulse function is symmetric.

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

$f\left(x\right)={x}^{2}\mathrm{cos}x-|\mathrm{sin}x|$

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

$⇒f\left(-x\right)={\left(-x\right)}^{2}\mathrm{cos}\left(-x\right)-|\mathrm{sin}\left(-x\right)|$

We know that :

${\left(-x\right)}^{2}={x}^{2};\phantom{\rule{1em}{0ex}}\mathrm{cos}\left(-x\right)=\mathrm{cos}x;\phantom{\rule{1em}{0ex}}|\mathrm{sin}\left(-x\right)|=|-\mathrm{sin}x|=|\mathrm{sin}x|$

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

$⇒f\left(-x\right)={\left(-x\right)}^{2}\mathrm{cos}\left(-x\right)-|\mathrm{sin}\left(-x\right)|={x}^{2}\mathrm{cos}x-|\mathrm{sin}x|=f\left(x\right)$

Hence, given function is an even function.

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

$f\left(x\right)=x{e}^{-{x}^{2}{\mathrm{tan}}^{2}x}$

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

$⇒f\left(-x\right)=\left(-x\right){e}^{-\left\{{\left(-x\right)}^{2}{\mathrm{tan}}^{2}\left(-x\right)\right\}}$

We know that :

${\left(-x\right)}^{2}={x}^{2};\phantom{\rule{1em}{0ex}}{\mathrm{tan}}^{2}\left(-x\right)={\left(-\mathrm{tan}x\right)}^{2}={\mathrm{tan}}^{2}x$

$⇒f\left(-x\right)=\left(-x\right){e}^{-\left\{{\left(-x\right)}^{2}\mathrm{tan}{}^{2}\left(-x\right)\right\}}=-x{e}^{-{x}^{2}\mathrm{tan}{}^{2}x}=-f\left(x\right)$

Hence, given function is an odd function.

how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
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
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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 biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
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
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
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
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x