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Even and odd functions are related to symmetry of functions. The symmetry of a function is visualized by the planar plot of a function, which may show symmetry with respect to either an axis (y-axis) or origin.

Since functions need not always be symmetric, they may neither be even nor be odd. The parity of a function i.e. whether it is even or odd is determined with certain algebraic algorithm. Further, symmetry of functions may change subsequent to mathematical operations.

Even functions

The values of even function at x=x and x=-x are same.

Even function
A function f(x) is said to be “even” if for every “x”, there exists “-x” in the domain of the function such that :

f - x = f x

An even function is symmetric about y-axis. If we consider the axis as a mirror, then the plot in first quadrant has its mirror image (bilaterally inverted) in second quadrant. Similarly, the plot in fourth quadrant has its mirror image (bilaterally inverted) in third quadrant.

Some examples of even functions are x 2 , | x | and cos x . In each case, we see that :

f - x = - x 2 = x 2 = f x

f - x = | - x | = | x | = f x

f - x = cos - x = cos x = f x

The right side is mirror image of left hand side and the left side is mirror image of right hand side of the curve.

Even functions

Examples of even functions.

It is important to see that if we rotate the curve by 180° about y-axis, then the appearance of the rotated curve is same as the original curve. We can state this alternatively as : if we rotate left hand side of the curve by 180° about y-axis, then we get the right hand curve and vice-versa.

Examples

Problem 1: Prove that the function f(x) is “even”, if

f x = x a x 1 a x + 1

Solution : For function being “even”, we need to prove that :

f - x = f x

Here,

f x = x a x 1 a x + 1 = x 1 a x 1 1 a x + 1

f x = x 1 a x a x 1 + a x a x = x 1 a x 1 + a x

f x = x a x 1 a x + 1 = f x

Problem 2: If an even function “f” is defined on the interval (-5,5), then find the real values for which

f x = f x + 1 x + 2

Solution : It is given that function “f” is even. Hence, arguments of the functions on two sides are related either as

x = x + 1 x + 2

or as :

x = x + 1 x + 2

From the first relation,

x 2 + x 1 = 0

x = - 1 ± 5 2

From the second relation,

x 2 + 3 x + 1 = 0

x = - 3 ± 5 2

We see that values are within the specified domain. Hence, all the four solutions satisfy the given equation.

Odd functions

The values of odd function at x=x and x=-x are equal in magnitude but opposite in sign.

Odd function
A function f(x) is said to be “odd” if for every “x”, there exists “-x” in the domain of the function such that :

f - x = - f x

An odd function is symmetric about origin of the coordinate system. The plot in first quadrant has its mirror image (bilaterally inverted) in third quadrant. Similarly, the plot in second quadrant has its mirror image (bilaterally inverted) in fourth quadrant.

Some examples of odd functions are : x , x 3 and sin x . In each case, we see that :

f - x = - x = - f x

f - x = - x 3 = - x 3 = - f x

f - x = sin - x = - sin x = - f x

The upper curve of these functions is exactly same as the lower curve across x-axis.

Odd functions

Examples of odd functions.

It is important to see that if we rotate the curve by 180° about origin, then the appearance of the rotated curve is same as the original curve. In other words, if we rotate right hand side of curve by 180° about origin, then we get left side of the curve. Further, it is interesting to note that we obtain left hand part of the plot of odd function in two steps : (i) drawing reflection (mirror image) of right hand plot about y-axis and (ii) drawing reflection (mirror image) of “reflection drawn in step 1” about x-axis.

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?
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
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
Mapo Reply
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
Mapo Reply
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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