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x = y n

Interchangeably, we write :

y = x 1 n = n x

If n is even integer, then x can not be negative. For n=2, we drop “n” from the notation and we write,

y = x

We extend this concept to function in which number “x” is substituted by any valid expression (algebraic, trigonometric, logaritmic etc). Some examples are :

y = x 2 + 3 x 5 y = log e x 2 + 3 x 5

We shall also include study of radical function which is part of rational form like :

y = 1 x 2 + 3 x - 5

y = { x - 1 x + 3 x 2 + 3 x - 5 }

Analysis of radical function

Analysis of root function is same as analysis of inequality of function. Because, radical function ultimately results in inequality. We make use of the fact that expression within the radical sign is non-negative. Here, we denote a radical function as :

f x = g x

As the expression under is non-negative,

g x 0

When radical function is part of a function defined in rational form, the radical function should not be zero. Let us consider a function as :

f x = 1 g x

As the radical is denominator of the rational expression, expression under radical sign is positive,

g x > 0

We have already worked with inequalities involving polynomial and rational functions. We shall restrict ourselves to few illustrations here.

Problem : Find domain of the function :

f x = { 1 - 1 - x 2 }

Solution : One radical (inner) is contained with another radical (outer). For the outer radical,

1 - 1 - x 2 0 1 - x 2 1

The term on each side of inequality is a positive quantity. Squaring each side does not change inequality,

1 - x 2 1 x 2 0

This quadratic inequality is true for all real x. Now, for inner radical

1 - x 2 0 1 + x 1 x 0

We multiply by -1 to change the sign of x in 1-x,

x + 1 x 1 0

Using sign rule :

x [ - 1,1 ]

Since, conditions corresponding to two radicals need to be fulfilled simultaneously, the domain of the given function is intersection of outer and inner radicals.

Domain of the function

Intersection of two domains.

Domain = [ - 1,1 ]

Problem : Find the domain of the function given by :

f x = x 14 x 11 + x 6 x 3 + x 2 + 1

Solution : Clearly, function is real for values of “x” for which expression within square root is a non negative number. We note that independent variable is raised to positive integers. The nature of each monomial depends on the value of x and nature of power. If x≥1, then monomial evaluates to higher value for higher power. If x lies between 0 and 1, then monomial evaluates to lower value for higher power. Further, a negative x yields negative value when raised to odd power and positive value when raised to even power. We shall use these properties to evaluate the expression for three different intervals of x.

x 14 x 11 + x 6 x 3 + x 2 + 1 0

We consider different intervals of values of expression for different values of “x”, which cover the complete interval of real numbers.

1: x 1

In this case, x a > x b , if a > b . Evaluating in groups,

x 14 x 11 + x 6 x 3 + x 2 + 1 > 0

2: 0 x < 1

In this case, x a < x b , if a > b . Rearranging in groups,

x 14 { x 11 x 6 + x 3 x 2 } + 1

Here, { x 11 x 6 + x 3 x 2 } is negative. Hence total expression is positive,

x 14 { x 11 x 6 + x 3 x 2 } + 1 > 0

3: x < 0

Rearranging in groups,

x 14 x 11 + x 6 x 3 + x 2 + 1

Here, x 14, x 6 and x 2 are positive and x 11 and x 3 is negative. Hence, total expression is positive,

x 14 x 11 + x 6 x 3 + x 2 + 1 > 0

We see that expression is positive for all values of “x”. Hence, domain of the function is :

Domain = R = ,

Exercise

Find solution of the rational inequality given by :

x + 1 x + 5 x - 3 0

Hint : Critical points are -5,-1 and 3. We need to exclude end corresponding to x=3 as denominator turns zero for this value.

[ - 5 , - 1 ] ( 3 , )

Find solution of the rational inequality given by :

8 x 2 + 16 x 51 2 x 3 x + 4 > 0

Hint : Critical points are -4,-3,3/2,5/2.

x - , - 4 - 3,3 / 2 5 / 2,

Find solution of the rational inequality given by :

x 2 + 4 x + 3 x 3 6 x 2 + 11 x 6 > 0

Hint : Factorize denominator as x 3 6 x 2 + 11 x 6 = x 1 x 2 x 3 . Critical points are -3,-1,1,2,3.

x - 3, - 1 1,2 3,

Find solution of the rational inequality given by :

2 x + 1 x 1 2 x 3 3 x 2 + 2 x 0

Hint : Factorize denominator as

2 x + 1 x 1 2 x 3 3 x 2 + 2 x = 2 x + 1 x 1 2 x x 1 x 2

Critical points are -1/2,1,1,0,1 and 2. We see that "1" is repeated odd times. Hence, we continue to assign alternating signs in accordance with wavy curve method. The solution of x for the inequality is :

- < x - 1 / 2 0 < x < 1 2 < x <

Questions & Answers

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
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
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
How can I make nanorobot?
Lily
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
how can I make nanorobot?
Lily
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
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
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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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