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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.


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

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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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