# 5.4 Monotonic functions  (Page 3/3)

 Page 3 / 3

$\text{Range}=\left[f\left({x}_{1}\right),f\left({x}_{2}\right)\right]$

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,

$\text{If}\phantom{\rule{1em}{0ex}}{x}_{1}<{x}_{2}\phantom{\rule{1em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}f\left({x}_{1}\right)\le f\left({x}_{2}\right)$

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,

$f\prime \left(x\right)\ge 0;\phantom{\rule{1em}{0ex}}\text{Equality sign holds for few points or a continuous section in X}$

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,

$\text{If}\phantom{\rule{1em}{0ex}}{x}_{1}<{x}_{2}\phantom{\rule{1em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}f\left({x}_{1}\right)>f\left({x}_{2}\right)$

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,

$⇒{x}_{1}^{2}>{x}_{2}^{2}$ $⇒f\left({x}_{1}\right)>f\left({x}_{2}\right)$

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,

$f\prime \left(x\right)\le 0;\phantom{\rule{1em}{0ex}}\text{Equality sign holds for points only - not on a continuous section in X}$

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,

$f\left(x\right)=\mathrm{sin}x$ $⇒f\prime \left(x\right)=\mathrm{cos}x$

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 :

$\text{Range}=\left[f\left({x}_{2}\right),f\left({x}_{1}\right)\right]$

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,

$\text{If}\phantom{\rule{1em}{0ex}}{x}_{1}<{x}_{2}\phantom{\rule{1em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}f\left({x}_{1}\right)\ge f\left({x}_{2}\right)$

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,

$f\prime \left(x\right)\le 0;\phantom{\rule{1em}{0ex}}\text{Equality sign holds for few points or a continuous section in X}$

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.

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
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 to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
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