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Range = [ f x 1 , f x 2 ]

We shall study this aspect of finding range in detail in a separate module.

Non-decreasing function or increasing

The successive value of function increases or remains constant as the value of the independent variable increases. In other words, the preceding values are less than or equal to successive values that follow. Mathematically,

If x 1 < x 2 then f x 1 f x 2

Non decreasing function

The successive function value is greater than or equal to previous value.

As f( x 1 )≤f( x 2 ) for all x 1 , x 2 ∈X, the difference “f(x+h) – f(x)” is non-negative for “h”, however small. This implies that the first derivative of function is non-negative. If we think of possibility, then we can realize that tangent to the function curve can be parallel to x-axis for a subset of X, while curve is increasing overall in the interval. It means that first derivative can be equal to zero points or sub-intervals in which it is increasing. Thus, for non-decreasing function,

f x 0 ; Equality sign holds for few points or a continuous section in X

For increasing function, if x 1 < x 2 , then f( x 1 ) ≤ f( x 2 ), for all x 1 , x 2 ∈X. This means that there may be same function values for different values of x. This is “many one” relation and as such function is not invertible in X.

Strictly decreasing function

The successive value of function decreases as the value of the independent variable increases. In other words, the preceding values are greater than successive values that follow. Mathematically,

If x 1 < x 2 then f x 1 > f x 2

Strictly decreasing function

The successive function value is less than previous value.

Problem : Determine monotonic nature of the function in the interval (-∞,0].

y = x 2

Solution : Let x 1 and x 2 belong to the interval [0,∞) such that x 1 < x 2 . Multiplying inequality with x 1 (a negative number) changes the nature of inequality :

x 1 2 > x 1 x 2

Multiplying inequality with x 2 (a negative number) changes the nature of inequality :

x 1 x 2 > x 2 2

Combining two inequalities,

x 1 2 > x 2 2 f x 1 > f x 2

Thus, given function is strictly decreasing in (-∞,0].

As f( x 1 )>f( x 2 ) for all x 1 , x 2 ∈X, the difference “f(x+h) – f(x)” is negative for “h”, however small. This implies that the first derivative of function is negative. If we think of possibility, then we can realize that tangent to the function curve can be parallel to x-axis for couple of x values, while curve is continuously decreasing in the interval. It means that first derivative can be equal to zero for few points in the interval in which it is strictly decreasing. Thus, for strictly decreasing function,

f x 0 ; Equality sign holds for points only - not on a continuous section in X

For strictly decreasing function, if x 1 < x 2 , then f( x 1 )>f( x 2 ), for all x 1 , x 2 ∈X. It means that all distinct x values correspond to distinct y values and vice-versa. Therefore, strictly decreasing function is one-one function i.e. a bijection and hence “invertible”. In other words, if a function has strict decreasing order, then it is invertible. Mathematically, we say that if f’(x) ≤ 0 (equality holding for points only); x∈X, then function is invertible in X. For example, consider sine function,

f x = sin x f x = cos x

We know that cosx is negative in the interval [π/2, 3π/2]. Hence sine function is a strictly decreasing function in [π/2, 3π/2]and is invertible. Recall though that inverse sine function is not defined in this interval, but in basic interval about origin [-π/2,π/2].

The order of a function provides an easy technique to determine range of a continuous function, corresponding to a given domain interval. For example, if domain of a continuously decreasing function, f(x), is [ x 1 , x 2 ], then the least value of the function is f( x 2 ) and greatest value of the function is f( x 1 ). Hence, range of the function is :

Range = [ f x 2 , f x 1 ]

We shall study this aspect of finding range in detail in a separate module.

Non-increasing function or decreasing

The successive value of function decreases or remains constant as the value of the independent variable increases. In other words, the preceding values are greater than or equal to successive values that follow. Mathematically,

If x 1 < x 2 then f x 1 f x 2

Non-increasing function or decreasing

The successive function value is less than or equal to previous value.

As f( x 1 )≥ f( x 2 ) for all x 1 , x 2 ∈X, the difference “f(x+h) – f(x)” is non-positive for “h”, however small. This implies that the first derivative of function is non-positive. If we think of possibility, then we can realize that tangent to the function curve can be parallel to x-axis for a subset of X, while curve is decreasing overall in the interval. It means that first derivative can be equal to zero at points or in sub-intervals in which it is decreasing. Thus, for non-decreasing function,

f x 0 ; Equality sign holds for few points or a continuous section in X

For decreasing function, if x 1 < x 2 , then f( x 1 ) ≥ f( x 2 ), for all x 1 , x 2 ∈X. This means that there may be same function values for different values of x. This is “many one” relation and as such function is not invertible in X.

Questions & Answers

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
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
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
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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