3.4 Rational function  (Page 4/6)

 Page 4 / 6

2. Find horizontal asymptote :

$f\left(x\right)=\frac{-x\left(x+1\right)}{{x}^{4}+16}\phantom{\rule{1em}{0ex}}$

The order of highest power term is 2 in numerator and 4 in denominator. Thus, n<m. Hence, x-axis is horizontal asymptote.

$y=0$

3. Find horizontal asymptote :

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

The order of highest power term is 3 in numerator and 1 in denominator. Here, n>m. Hence, there is no horizontal asymptote.

Slant asymptotes

Slant asymptote is a line that the graph approaches. This line is neither vertical nor horizontal. A rational function has a slant or oblique asymptote when order of numerator (n) is greater than order of denominator (m).

The equation of slant asymptote is obtained by dividing numerator polynomial by denominator polynomial. The quotient of division is equation of asymptote. Clearly, asymptote is a straight line. As such, quotient should be a linear expression. The requirement that asymptote is a straight line implies that the order of numerator polynomial is higher than order of denominator polynomial by 1 i.e. n=m+1.

In the nutshell, slant asymptote exists when n=m+1. The slant asymptote is obtained by dividing numerator and denominator. We neglect remainder. The equation of the slant asymptote is given by quotient equated to “y”.

Find slant asymptote :

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

Division here yields quotient as “x-1”. Hence, equation of slant asymptote is :

$y=x-1$

X-intercepts

The x-intercepts are also known as zeroes of function or real roots of corresponding equation when function is equated to zero. Since function is many-one, there can be more than one x-intercept. On graphs, x-intercepts are points on x-axis, where graph intersects it. Thus, x-intercepts are x-values where function value becomes zero.

$f\left(x\right)=0$

In the case of rational function, x-intercepts exist when numerator turns zero. In other words, x-intercepts are x-values for which numerator of the function turns zero.

$g\left(x\right)=\frac{f\left(x\right)}{h\left(x\right)}=0$ $g\left(x\right)=0\phantom{\rule{1em}{0ex}}⇒f\left(x\right)=0$

We determine x-intercepts by solving equation formed by equating function to zero. It is hepful to know that real polynomial of odd degree has a real root and, therefore, at least one x-intercept. In the case of rational function, the function is not defined for values of x when denominator turns zero. It means that there will be no x-intercept corresponding to linear factor which is common to denominator. Consider the function given here :

$h\left(x\right)=\frac{\left(x-1\right)\left(x+2\right)}{{\left(x-1\right)}^{2}\left(x+1\right)}$

Equating numerator to zero, we have :

$⇒\left(x-1\right)\left(x+2\right)=0$ $⇒x=1,-2$

But (x-1) is also linear factor in denominator. It means that point x=1 is a singularity. Hence, x-intercept is only x=-2 as function is not defined at x=1.

Find x-intercepts of reciprocal function :

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

Here numerator is 1 and can not be zero. Thus, reciprocal function does not have x-intercepts.

Find x-intercepts of function given by :

$\frac{{x}^{2}-3x+2}{{x}^{2}-2x-3}$

Factorizing, we have :

$\frac{{x}^{2}-3x+2}{{x}^{2}-2x-3}=\frac{\left(x-1\right)\left(x-2\right)}{\left(x+1\right)\left(x-3\right)}$

Equating numerator to zero, we have :

$⇒\left(x-1\right)\left(x-2\right)=0$ $⇒x=1\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}2$

Thus, x-intercepts are 1 and 2.

Y-intercepts

This is function or y value when x is zero. Functions are many-one relation. Thus, there can be only one y-intercept. The y-intercept is calculated as :

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
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
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
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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
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