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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

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Eke Reply
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Missy Reply
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Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Preparation and Applications of Nanomaterial for Drug Delivery
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has a lot of application modern world
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what about nanotechnology for water purification
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please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
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yes that's correct
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I think
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Nasa has use it in the 60's, copper as water purification in the moon travel.
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what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
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How we are making nano material?
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
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The nanotechnology is as new science, to scale nanometric
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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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