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We have seen that a function is a special relation. In the same sense, real function is a special function. The special about real function is that its domain and range are subsets of real numbers “R”. In mathematics, we deal with functions all the time – but with a difference. We drop the formal notation, which involves its name, specifications of domain and co-domain, direction of relation etc. Rather, we work with the rule alone. For example,

f x = x 2 + 2 x + 3

This simplification is based on the fact that domain, co-domain and range are subsets of real numbers. In case, these sets have some specific intervals other than “R” itself, then we mention the same with a semicolon (;) or a comma(,) or with a combination of them :

f x = x + 1 2 1 ; x < - 2 , x 0

Note that the interval “ x < - 2 , x 0 ” specifies a subset of real number and defines the domain of function. In general, co-domain of real function is “R”. In some cases, we specify domain, which involves exclusion of certain value(s), like :

f x = 1 1 x , x 1

This means that domain of the function is R { 1 } . Further, we use a variety of ways to denote a subset of real numbers for domain and range. Some of the examples are :

  • x > 1 : denotes subset of real number greater than “1”.
  • R { 0 , 1 } denotes subset of real number that excludes integers “0” and “1”.
  • 1 < x < 2 : denotes subset of real number between “1” and “2” excluding end points.
  • ( 1,2 ] : denotes subset of real number between “1” and “2” excluding end point “1”, but including end point “2”.

Further, we may emphasize the meaning of following inequalities of real numbers as the same will be used frequently for denoting important segment of real number line :

  • Positive number means x>0 (excludes “0”).
  • Negative number means x<0 (excludes “0”).
  • Non - negative number means x ≥ 0 (includes “0”).
  • Non – positive number means x ≤ 0 (includes “0”).

Domain of real function

Domain of real function is a subset of “R” such that rule “f(x)” is real. This concept is simple. We need to critically examine the given function and evaluate interval of “x” for which “f(x)” is real.

In this module, we shall restrict ourselves to algebraic functions. We determine domain of algebraic function for being real in the light of following facts :

  • If the function has rational form p(x)/q(x), then denominator q(x) ≠ 0.
  • The term √x is a positive number, where x>=0.
  • The expression under even root should be non-negative. For example the function x 2 + 3 x - 2 to be real, x 2 + 3 x 2 0 .
  • The expression under even root in the denominator of a function should be positive number. For example the function 1 x 2 + 3 x - 2 to be real, x 2 + 3 x - 2 > 0 . Note that zero value of expression is not permitted in the denominator.

Here, we shall work with few examples as illustration for determining domain of real function.

Examples

Problem 1 : A function is given by :

f x = 1 x + 1

Determine its domain set.

Solution : The function, in the form of rational expression, needs to be checked for its denominator. The denominator should not evaluate to zero as “a/0” form is undefined. For given function in the question, the denominator evaluates to zero when,

Questions & Answers

Application of nanotechnology in medicine
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
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
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
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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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