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Two numbers are equal if they are same number. Two variables are equal if they represent same number. Following these connotations, two functions are equal if they are same function. But, very concept of “equal” or “identical” functions indicates that there is more than one way to represent a function. In other words, the question of equality of two functions arises when two function forms yield same values. There are few such occurrences in mathematics. This arises primarily because we have alternate ways to represent a mathematical entity. Consider, for example, modulus function. There are two equivalent expressions :

f x = | x | g x = x 2

These two function forms yield same values for all real values of x. As such, these two functions f(x) and g(x) are equal functions. On the other hand, there are equivalent forms, which represent equal values but not for all values of x in the domains of two definition. Consider, for example,

f x = 2 log e x g x = log e x 2

The logarithmic function f(x) is defined for x>0. This means its domain is (0, ∞). For logarithmic function, g(x),

x 2 > 0

This inequality is true for all values of x except x=0. It means domain of g(x) is R-{0}. Clearly, domains of two functions are not equal. For a value x = -1, g(x) yields a value while f(x) is not defined for this value of x. Two equations, therefore, are not equal. However, two functions are equal if we limit our consideration for domain limited to the intersection of two domains. Hence,

f x = g x ; x 0,

There is yet another possibility. Two equivalents forms have same domains, but yield different set of values. In such case also, two functions are not equal. Consider the example given here.

Problem : Determine whether f(x) and g(x) are identical functions?

f x = x g x = x 2

Solution : Here, f(x) is defined for all values of x and its domain is R. On the other hand, domain of g(x) is also R as square of x is always non-negative. However, square root of a number is non-negative. Therefore, two function forms are not equivalent as f(x) is real, whereas is g(x) is non-negative and is a subset of R. Thus, range of f(x) is R and range of g(x) is (0,∞). Clearly, two given functions are not equal.

In the nutshell, two equivalent function forms are equal if their domain, range and function values are equal.

Definition of equal functions

Two functions f(x) and g(x) are equal functions, if :

(i) Domain of f (x) = Domain of g(x) = X

(ii) f(x) = g(x) for all x X

Equal functions are also known as identical functions. Above two conditions are sufficient for two functions to be equal. Since second condition means that values of functions are equal for every x in the domain, it is guaranteed that range of two functions are equal.

Range of f (x) = Range of g(x) = Y

Examples

Problem : Determine whether f(x) and g(x) are identical functions.

f x = x x 2 g x = 1 x

Solution : Two function forms are equivalent as f(x) is reduced to g(x) on simplification. Now, expression of f(x) is defined for all values of x except x=0. Thus, domain of f(x) is R-{0}. On the other hand, domain of reciprocal function g(x) is also R-{0}. Clearly, two given functions are equal.

Problem : 3. Determine whether f(x) and g(x) are identical functions.

f x = log e x log e x 2 + 1 g x = log e x 1 + x 2

Solution :

Two function forms are equivalent as f(x) is changed to g(x) and vice-versa on simplification. Now, f(x) is defined for

x > 0 and x 2 + 1 > 0

But x 2 is always positive. Hence, domain of f(x) is (0, ∞). On the other hand, g(x) is defined for :

x 1 + x 2 > 0 x > 0

Thus, domain of g(x) is also (0, ∞). Hence, two functions are identical.

Problem : Determine domains for which two functions are equal.

f x = log x - log x - 1 g x = log x x - 1

Solution : Two function forms are equivalent as f(x) is changed to g(x) and vice-versa on simplification. Now, f(x) is defined for

x > 0 and x - 1 > 0 x > 0 and x > 1

Hence, domain of f(x) is intersection of two intervals (1, ∞). On the other hand, g(x) is defined for :

x x 1 > 0

Critical points are 0 and 1. Using sign rule for rational function, the domain of g(x) is values of x satisfying above inequality :

, 0 1,

Clearly, two domains are not equal. Note that there is no restriction on the range of the functions. Therefore, two functions are equal in the restricted domain which is intersection of two domains.

Domain = 1,

Acknowledgment

Author wishes to thank Ms. Aditi Singh, New Delhi for her valuable suggestions on the topic.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
revolt
da
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
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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