Equal functions

 Page 1 / 1

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 :

$f\left(x\right)=|x|$ $g\left(x\right)=\sqrt{{x}^{2}}$

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,

$f\left(x\right)=2{\mathrm{log}}_{e}x$ $g\left(x\right)={\mathrm{log}}_{e}{x}^{2}$

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,

$f\left(x\right)=g\left(x\right);\phantom{\rule{1em}{0ex}}x\in \left(0,\infty \right)$

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?

$f\left(x\right)=x$ $g\left(x\right)=\sqrt{{x}^{2}}$

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 :

$\text{(i) Domain of f (x) = Domain of g(x) = X}$

$\text{(ii) f(x) = g(x) for all}\phantom{\rule{1em}{0ex}}x\in X$

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.

$\text{Range of f (x) = Range of g(x) = Y}$

Examples

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

$f\left(x\right)=\frac{x}{{x}^{2}}$ $g\left(x\right)=\frac{1}{x}$

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.

$f\left(x\right)={\mathrm{log}}_{e}x-{\mathrm{log}}_{e}\left({x}^{2}+1\right)$ $g\left(x\right)=\mathrm{log}{}_{e}\left(\frac{x}{1+{x}^{2}}\right)$

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

$x>0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{x}^{2}+1>0$

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

$\frac{x}{1+{x}^{2}}>0$ $x>0$

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

Problem : Determine domains for which two functions are equal.

$f\left(x\right)=\mathrm{log}x-\mathrm{log}\left(x-1\right)$ $g\left(x\right)=\mathrm{log}\left(\frac{x}{x-1}\right)$

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

$x>0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}x-1>0$ $x>0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}x>1$

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

$\frac{x}{x-1}>0$

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

$\left(-\infty ,0\right)\cup \left(1,\infty \right)$

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.

$\text{Domain}=\left(1,\infty \right)$

Acknowledgment

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

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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
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
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
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