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  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.

y 1 y 0

y 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 :

Range of a function

Range of the function is equal to intersection of two intervals.

0 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 2 x + 1

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

sin x 2 + x y + y 2 = x y

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

x 0 , - , - 1 , , 0 X ( - )

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 " - 1 " is undefined, because it is not certain whether the expression will evaluate to "-1" or "1". On the other hand, expression " 1 " is defined because we are sure that the expression will evaluate to "1", however large is infinity.

Defined in some contexts

0 0 = undefined or 1

0 = undefined or 1

1 = undefined or 1

0 X = 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 = , where “x” is real.

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

- < x < or - ,

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.

± x =

- ± x = -

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.

+ =

3: Difference of two infinities is undefined.

= 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 :

X =

- X = -

- X - =

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

x X = , if x > 0

x X = - , if x < 0

x X = 0, if x = 0

6: Division of infinity by infinity is not defined.

= Undefined

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

x = 0, if 0 < x < 1

x = 0, if x = 0

x = , if x > 1

8: An infinity raised to infinity is defined.

=

Questions & Answers

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 to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
Devang Reply
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
Abhijith Reply
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?
s. Reply
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 ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
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
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Period of sin^6 3x+ cos^6 3x
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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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