# 5.6 Rational functions  (Page 11/16)

 Page 11 / 16

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

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: $x\to ±\infty ,\phantom{\rule{0.2em}{0ex}}f\left(x\right)\to \frac{1}{3}$

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.

the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al