# 5.1 Even and odd functions  (Page 2/3)

 Page 2 / 3

## Examples

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

$f\left(x\right)={\mathrm{log}}_{e}\left\{x+\sqrt{\left({x}^{2}+1\right)}\right\}$

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

$⇒f\left(-x\right)={\mathrm{log}}_{e}\left[-x+\sqrt{\left\{}{\left(-x\right)}^{2}+1\right\}\right]={\mathrm{log}}_{e}\left\{-x+\sqrt{\left({x}^{2}+1\right)}\right\}$

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,

$⇒f\left(-x\right)={\mathrm{log}}_{e}\left[\frac{\left\{-x+\sqrt{\left({x}^{2}+1\right)}\right\}X\left\{x+\sqrt{\left({x}^{2}+1\right)}\right\}}{\left\{x+\sqrt{\left({x}^{2}+1\right)}\right\}}\right]={\mathrm{log}}_{e}\left[\frac{-{x}^{2}+{x}^{2}+1}{\left\{x+\sqrt{\left({x}^{2}+1\right)}\right\}}\right]$

$⇒f\left(-x\right)={\mathrm{log}}_{e}1-{\mathrm{log}}_{e}\left\{x+\sqrt{\left({x}^{2}+1\right)}\right\}=-{\mathrm{log}}_{e}\left\{x+\sqrt{\left({x}^{2}+1\right)}\right\}=-f\left(x\right)$

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),

$f\left(-x\right)=\mathrm{sin}\left(-x\right)+\mathrm{cos}\left(-x\right)=-\mathrm{sin}x+\mathrm{cos}x$

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,

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

But, “g” is an odd function. Hence,

$⇒g\left(-x\right)=-g\left(x\right)$

Combining two equations,

$⇒fog\left(-x\right)=f\left(-g\left(x\right)\right)$

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

$⇒fog\left(-x\right)==f\left(-g\left(x\right)\right)=f\left(g\left(x\right)\right)=fog\left(x\right)$

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”,

$⇒h\left(-x\right)=\frac{f\left(-x\right)}{g\left(-x\right)}$

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

$⇒h\left(-x\right)=\frac{f\left(x\right)}{-g\left(x\right)}=-h\left(x\right)$

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.

what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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?
what is a peer
What is meant by 'nano scale'?
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
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
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 By By   By  By   