# Domain and range  (Page 4/4)

 Page 4 / 4

1. Total expression within the square root as a whole is non-negative number as square root of a negative number is not a real number.
2. For positive value of "y" in the numerator, the denominator is non-negative as square root of a negative number is not a real number.
3. The denominator does not evaluate to zero. The form “y/0” is undefined.

For the first requirement, the expression in the square root should be greater than or equal to zero i.e non-negative number.

$\frac{y}{1-y}\ge 0$

$⇒y\ge 0$

Further, denominator of the expression “1-y” is non-negative. Also, “1-y” is not equal to zero. Combining two requirements, the expression is a positive number :

$1-y>0$

$⇒y<1$

Combining two intervals i.e. intersection of two intervals, we have the range of the function as :

$⇒0\le y<1$

## Classification of functions

In mathematics, we deal with specific real functions, which are characterized by specific domain, range and rules. Some of the familiar functions are polynomial, rational, irrational, trigonometric, exponential, logarithmic functions and piece wise defined functions etc. These functions are further combined to form more complex function following certain definition or rule so that function is meaningful for real values.

There are varieties of functions. These functions are broadly classified under three headings :

• Algebraic functions : polynomial, rational and irrational functions
• Transcendental functions : trigonometric, inverse trigonometric, exponential and logarithmic functions
• Piece wise defined functions : modulus, greatest integer, least integer, fraction part functions and other specific piece wise definitions

Polynomial function is further classified based on (i) numbers of terms eg. monomial, binomial, trinomial etc. (ii) numbers of variables involved eg. function in one or two variables and (iii) degree of the polynomial eg. linear, quadratic, cubic, bi-quadratic etc.

Generally we deal with function expressions in one variable in which dependent variable (y) is explicitly related to independent variable (x). Such functions are called explicit function.

$y={x}^{2}-2x+1$

On the other hand, there are function rules in which “y” is not explicitly related to “x”. Such functions are called implicit functions.

$\mathrm{sin}\left({x}^{2}+xy+{y}^{2}\right)=xy$

Further, we also use properties of function like periodicity (repetition of function values at regular intervals of independent variable) and polarity (odd or even) to characterize a function. For this reason, we sometimes name a function like periodic, non-periodic, aperiodic, odd, even or equal function etc.

## Important concepts

In this section, we discuss few important concepts, which are frequently used in determining domain and range.

## Defined and undefined expressions

Defined expressions are meaningful and unambiguous. On the contrary, undefined expressions are not meaningful. Most of the undefined expressions results from combination of zeros and infinity in various ways. There is, however, no unanimity about “undefined” values among mathematicians. Hence, we shall present two lists – one list which is undefined in all contexts and another list which may be defined in certain context. We consider this later list as defined expressions, unless otherwise stated.

Undefined in all contexts

$\frac{x}{0},\phantom{\rule{1em}{0ex}}\infty -\infty ,\phantom{\rule{1em}{0ex}}{\left(-1\right)}^{\infty },\phantom{\rule{1em}{0ex}}\frac{\infty }{\infty },\phantom{\rule{1em}{0ex}}0X\left(-\infty \right)$

We should understand here that an expression is undefined when it can not be interpreted. The important point is that it has got nothing to do with the magnitude of quantity. Emphatically, infinity is not undefined. We shall discuss this aspect subsequently.

Note that " ${\left(-1\right)}^{\infty }$ " is undefined, because it is not certain whether the expression will evaluate to "-1" or "1". On the other hand, expression " ${\left(1\right)}^{\infty }$ " is defined because we are sure that the expression will evaluate to "1", however large is infinity.

Defined in some contexts

${0}^{0}=\text{undefined or 1}$

${\infty }^{0}=\text{undefined or 1}$

${1}^{\infty }=\text{undefined or 1}$

$0X\infty =\text{undefined or 0}$

For our purpose, unless otherwise stated, we shall consider later set as defined.

## Infinity

Infinity is not a member of real number set “R”. Strictly we can not write infinity like :

$x=\infty ,\phantom{\rule{1em}{0ex}}\text{where “x” is real.}$

For this reason, the interval of real number set is defined in terms of infinity without equality as :

$-\infty

We may emphasize that infinity by itself is “unbounded” – not undefined. What it means that we can interpret infinity – even though its value is not known. We can say it is very large number (either positive or negative as the case be), but we can not interpret undefined values at all.

It is easy to interpret operations with infinity. We need to only keep the meaning of infinity as a very large number in focus. Various operations, involving infinity are presented here :

1: The plus or minus infinity is not changed by adding or subtracting real number.

$\infty ±x=\infty$

$-\infty ±x=-\infty$

Above results are on expected line. Addition or subtraction of finite values will only yield a large number. It is so because infinity can be greater than a large value that we might conceive.

2: Addition of two infinities is infinity.

$\infty +\infty =\infty$

3: Difference of two infinities is undefined.

$\infty -\infty =\text{Undefined}$

Addition of two infinities is definitely a very large number. We are, however, not sure about their difference. The difference of two infinities can either be small or large. It depends on the relative "largeness" of two infinities. Hence, difference of two infinities is "undefined".

4: Product of two infinities are inferred as :

$\infty X\infty =\infty$

$-\infty X\infty =-\infty$

$-\infty X-\infty =\infty$

5: Product of infinity with a real number “x” is given as :

$xX\infty =\infty ,\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{1em}{0ex}}x>0$

$xX\infty =-\infty ,\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{1em}{0ex}}x<0$

$xX\infty =0,\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{1em}{0ex}}x=0$

6: Division of infinity by infinity is not defined.

$\frac{\infty }{\infty }=\text{Undefined}$

7: A real number, “x”, raised to infinity

${x}^{\infty }=0,\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{1em}{0ex}}0

${x}^{\infty }=0,\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{1em}{0ex}}x=0$

${x}^{\infty }=\infty ,\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{1em}{0ex}}x>1$

8: An infinity raised to infinity is defined.

${\infty }^{\infty }=\infty$

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
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
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
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