# 3.6 Rational functions  (Page 11/16)

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

Local behavior: $\text{\hspace{0.17em}}x\to -{\frac{1}{2}}^{+},f\left(x\right)\to -\infty ,x\to -{\frac{1}{2}}^{-},f\left(x\right)\to \infty \text{\hspace{0.17em}}$

End behavior: $\text{\hspace{0.17em}}x\to ±\infty ,f\left(x\right)\to \frac{1}{2}$

$f\left(x\right)=\frac{2x}{x-6}$

$f\left(x\right)=\frac{-2x}{x-6}$

Local behavior: $\text{\hspace{0.17em}}x\to {6}^{+},f\left(x\right)\to -\infty ,x\to {6}^{-},f\left(x\right)\to \infty ,\text{\hspace{0.17em}}$ End behavior: $\text{\hspace{0.17em}}x\to ±\infty ,f\left(x\right)\to -2$

$f\left(x\right)=\frac{{x}^{2}-4x+3}{{x}^{2}-4x-5}$

$f\left(x\right)=\frac{2{x}^{2}-32}{6{x}^{2}+13x-5}$

Local behavior: $\text{\hspace{0.17em}}x\to -{\frac{1}{3}}^{+},f\left(x\right)\to \infty ,x\to -{\frac{1}{3}}^{-},\text{\hspace{0.17em}}$ $f\left(x\right)\to -\infty ,x\to {\frac{5}{2}}^{-},f\left(x\right)\to \infty ,x\to {\frac{5}{2}}^{+}$ , $f\left(x\right)\to -\infty$

End behavior: $x\to ±\infty ,$ $f\left(x\right)\to \frac{1}{3}$

For the following exercises, find the slant asymptote of the functions.

$f\left(x\right)=\frac{24{x}^{2}+6x}{2x+1}$

$f\left(x\right)=\frac{4{x}^{2}-10}{2x-4}$

$y=2x+4$

$f\left(x\right)=\frac{81{x}^{2}-18}{3x-2}$

$f\left(x\right)=\frac{6{x}^{3}-5x}{3{x}^{2}+4}$

$y=2x$

$f\left(x\right)=\frac{{x}^{2}+5x+4}{x-1}$

## Graphical

For the following exercises, use the given transformation to graph the function. Note the vertical and horizontal asymptotes.

The reciprocal function shifted up two units.

The reciprocal function shifted down one unit and left three units.

The reciprocal squared function shifted to the right 2 units.

The reciprocal squared function shifted down 2 units and right 1 unit.

For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph.

$p\left(x\right)=\frac{2x-3}{x+4}$

$q\left(x\right)=\frac{x-5}{3x-1}$

$s\left(x\right)=\frac{4}{{\left(x-2\right)}^{2}}$

$r\left(x\right)=\frac{5}{{\left(x+1\right)}^{2}}$

$f\left(x\right)=\frac{3{x}^{2}-14x-5}{3{x}^{2}+8x-16}$

$g\left(x\right)=\frac{2{x}^{2}+7x-15}{3{x}^{2}-14+15}$

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

$b\left(x\right)=\frac{{x}^{2}-x-6}{{x}^{2}-4}$

$k\left(x\right)=\frac{2{x}^{2}-3x-20}{x-5}$

$w\left(x\right)=\frac{\left(x-1\right)\left(x+3\right)\left(x-5\right)}{{\left(x+2\right)}^{2}\left(x-4\right)}$

$z\left(x\right)=\frac{{\left(x+2\right)}^{2}\left(x-5\right)}{\left(x-3\right)\left(x+1\right)\left(x+4\right)}$

For the following exercises, write an equation for a rational function with the given characteristics.

Vertical asymptotes at $\text{\hspace{0.17em}}x=5\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=-5,\text{\hspace{0.17em}}$ x -intercepts at $\text{\hspace{0.17em}}\left(2,0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(-1,0\right),\text{\hspace{0.17em}}$ y -intercept at $\text{\hspace{0.17em}}\left(0,4\right)$

$y=50\frac{{x}^{2}-x-2}{{x}^{2}-25}$

Vertical asymptotes at $\text{\hspace{0.17em}}x=-4\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=-1,\text{\hspace{0.17em}}$ x- intercepts at $\text{\hspace{0.17em}}\left(1,0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(5,0\right),\text{\hspace{0.17em}}$ y- intercept at $\text{\hspace{0.17em}}\left(0,7\right)$

Vertical asymptotes at $\text{\hspace{0.17em}}x=-4\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=-5,\text{\hspace{0.17em}}$ x -intercepts at $\text{\hspace{0.17em}}\left(4,0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(-6,0\right),\text{\hspace{0.17em}}$ Horizontal asymptote at $\text{\hspace{0.17em}}y=7$

$y=7\frac{{x}^{2}+2x-24}{{x}^{2}+9x+20}$

Vertical asymptotes at $\text{\hspace{0.17em}}x=-3\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=6,\text{\hspace{0.17em}}$ x -intercepts at $\text{\hspace{0.17em}}\left(-2,0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(1,0\right),\text{\hspace{0.17em}}$ Horizontal asymptote at $\text{\hspace{0.17em}}y=-2$

Vertical asymptote at $\text{\hspace{0.17em}}x=-1,\text{\hspace{0.17em}}$ Double zero at $\text{\hspace{0.17em}}x=2,\text{\hspace{0.17em}}$ y -intercept at $\text{\hspace{0.17em}}\left(0,2\right)$

$y=\frac{1}{2}\frac{{x}^{2}-4x+4}{x+1}$

Vertical asymptote at $\text{\hspace{0.17em}}x=3,\text{\hspace{0.17em}}$ Double zero at $\text{\hspace{0.17em}}x=1,\text{\hspace{0.17em}}$ y -intercept at $\text{\hspace{0.17em}}\left(0,4\right)$

For the following exercises, use the graphs to write an equation for the function.

$y=4\frac{x-3}{{x}^{2}-x-12}$

$y=-9\frac{x-2}{{x}^{2}-9}$

$y=\frac{1}{3}\frac{{x}^{2}+x-6}{x-1}$

$y=-6\frac{{\left(x-1\right)}^{2}}{\left(x+3\right){\left(x-2\right)}^{2}}$

## Numeric

For the following exercises, make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote

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

 $x$ 2.01 2.001 2.0001 1.99 1.999 $y$ 100 1,000 10,000 –100 –1,000
$x$ 10 100 1,000 10,000 100,000
$y$ .125 .0102 .001 .0001 .00001

Vertical asymptote $\text{\hspace{0.17em}}x=2,\text{\hspace{0.17em}}$ Horizontal asymptote $\text{\hspace{0.17em}}y=0$

$f\left(x\right)=\frac{x}{x-3}$

$f\left(x\right)=\frac{2x}{x+4}$

 $x$ –4.1 –4.01 –4.001 –3.99 –3.999 $y$ 82 802 8,002 –798 –7998
 $x$ 10 100 1,000 10,000 100,000 $y$ 1.4286 1.9331 1.992 1.9992 1.999992

Vertical asymptote $\text{\hspace{0.17em}}x=-4,\text{\hspace{0.17em}}$ Horizontal asymptote $\text{\hspace{0.17em}}y=2$

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

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

 $x$ –.9 –.99 –.999 –1.1 –1.01 $y$ 81 9,801 998,001 121 10,201
 $x$ 10 100 1,000 10,000 100,000 $y$ 0.82645 0.9803 .998 .9998

Vertical asymptote $\text{\hspace{0.17em}}x=-1,\text{\hspace{0.17em}}$ Horizontal asymptote $\text{\hspace{0.17em}}y=1$

## Technology

For the following exercises, use a calculator to graph $\text{\hspace{0.17em}}f\left(x\right).\text{\hspace{0.17em}}$ Use the graph to solve $\text{\hspace{0.17em}}f\left(x\right)>0.$

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

$f\left(x\right)=\frac{4}{2x-3}$

$\left(\frac{3}{2},\infty \right)$

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

$f\left(x\right)=\frac{x+2}{\left(x-1\right)\left(x-4\right)}$

$\left(-2,1\right)\cup \left(4,\infty \right)$

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

## Extensions

For the following exercises, identify the removable discontinuity.

$f\left(x\right)=\frac{{x}^{2}-4}{x-2}$

$\left(2,4\right)$

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

$f\left(x\right)=\frac{{x}^{2}+x-6}{x-2}$

$\left(2,5\right)$

$f\left(x\right)=\frac{2{x}^{2}+5x-3}{x+3}$

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

$\left(–1,\text{1}\right)$

## Real-world applications

For the following exercises, express a rational function that describes the situation.

A large mixing tank currently contains 200 gallons of water, into which 10 pounds of sugar have been mixed. A tap will open, pouring 10 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of 3 pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ minutes.

A large mixing tank currently contains 300 gallons of water, into which 8 pounds of sugar have been mixed. A tap will open, pouring 20 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of 2 pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ minutes.

$C\left(t\right)=\frac{8+2t}{300+20t}$

For the following exercises, use the given rational function to answer the question.

The concentration $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ of a drug in a patient’s bloodstream $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ hours after injection in given by $\text{\hspace{0.17em}}C\left(t\right)=\frac{2t}{3+{t}^{2}}.\text{\hspace{0.17em}}$ What happens to the concentration of the drug as $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ increases?

The concentration $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ of a drug in a patient’s bloodstream $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ hours after injection is given by $\text{\hspace{0.17em}}C\left(t\right)=\frac{100t}{2{t}^{2}+75}.\text{\hspace{0.17em}}$ Use a calculator to approximate the time when the concentration is highest.

For the following exercises, construct a rational function that will help solve the problem. Then, use a calculator to answer the question.

An open box with a square base is to have a volume of 108 cubic inches. Find the dimensions of the box that will have minimum surface area. Let $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ = length of the side of the base.

A rectangular box with a square base is to have a volume of 20 cubic feet. The material for the base costs 30 cents/ square foot. The material for the sides costs 10 cents/square foot. The material for the top costs 20 cents/square foot. Determine the dimensions that will yield minimum cost. Let $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ = length of the side of the base.

$A\left(x\right)=50{x}^{2}+\frac{800}{x}.\text{\hspace{0.17em}}$ 2 by 2 by 5 feet.

A right circular cylinder has volume of 100 cubic inches. Find the radius and height that will yield minimum surface area. Let $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ = radius.

A right circular cylinder with no top has a volume of 50 cubic meters. Find the radius that will yield minimum surface area. Let $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ = radius.

$A\left(x\right)=\pi {x}^{2}+\frac{100}{x}.\text{\hspace{0.17em}}$ Radius = 2.52 meters.

A right circular cylinder is to have a volume of 40 cubic inches. It costs 4 cents/square inch to construct the top and bottom and 1 cent/square inch to construct the rest of the cylinder. Find the radius to yield minimum cost. Let $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ = radius.

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
Do somebody tell me a best nano engineering book for beginners?
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
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
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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