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Odd function plot

Odd function as two successive mirror images


Problem 3: Determine whether the function f(x) is “odd” function, where :

f x = log e { x + x 2 + 1 }

Solution : In order to determine the nature of function with respect to even or odd, we check for f(-x). Here,

f - x = log e [ - x + { - x 2 + 1 } ] = log e { - x + x 2 + 1 }

The expression on the right hand side can not be explicitly interpreted whether it equals to f(x) or not. Therefore, we rationalize the expression of logarithmic function,

f - x = log e [ { - x + x 2 + 1 } X { x + x 2 + 1 } { x + x 2 + 1 } ] = log e [ - x 2 + x 2 + 1 { x + x 2 + 1 } ]

f - x = log e 1 log e { x + x 2 + 1 } = log e { x + x 2 + 1 } = f x

Hence, given function is an “odd” function.

Problem 4: Determine whether sinx + cosx is an even or odd function?

Solution : In order to check the nature of the function, we evaluate f(-x),

f x = sin - x + cos - x = - sin x + cos x

The resulting function is neither equal to f(x) nor equal to “-f(x)”. Hence, the given function is neither an even nor an odd function.

Mathematical operations and nature of function

It is easy to find the nature of function resulting from mathematical operations, provided we know the nature of operand functions. As already discussed, we check for following possibilities :

  • If f(-x) = f(x), then f(x) is even.
  • If f(-x) = -f(x), then f(x) is odd.
  • If above conditions are not met, then f(x) is neither even nor odd.

Based on above algorithm, we can determine the nature of resulting function. For example, let us determine the nature of "fog" function when “f” is an even and “g” is an odd function. By definition,

f o g - x = f g - x

But, “g” is an odd function. Hence,

g - x = - g x

Combining two equations,

f o g - x = f - g x

It is given that “f” is even function. Therefore, f(-x) = f(x). Hence,

f o g - x = = f - g x = f g x = f o g x

Therefore, resulting “fog” function is even function.

The nature of resulting function subsequent to various mathematical operations is tabulated here for reference :

------------------------------------------------------------------------------------ f(x) g(x) f(x) ± g(x) f(x) g(x) f(x)/g(x), g(x)≠0 fog(x)------------------------------------------------------------------------------------ odd odd odd even even oddodd even Neither odd odd even even even even even even even------------------------------------------------------------------------------------

We should emphasize here that we need not memorize this table. We can always carry out particular operation and determine whether a particular operation results in even, odd or neither of two function types. We shall work with a division operation here to illustrate the point. Let f(x) and g(x) be even and odd functions respectively. Let h(x) = f(x)/g(x). We now substitute “x” by “-x”,

h - x = f - x g - x

But f(x) is an even function. Hence, f(-x) = f(x). Further as g(x) is an odd function, g(-x) = - g(x).

h - x = f x - g x = - h x

Thus, the division, here, results in an odd function.

There is an useful parallel here to remember the results of multiplication and division operations. If we consider even as "plus (+)" and odd as "minus (-)", then the resulting function is same as that resulting from multiplication or division of plus and minus numbers. Product of even (plus) and odd (minus) is minus(odd). Product of odd (minus) and odd (minus) is plus (even). Similarly, division of odd (minus) by even (plus) is minus (odd) and so on.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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