# 5.6 Rational functions  (Page 3/16)

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## Rational function

A rational function    is a function that can be written as the quotient of two polynomial functions

$f\left(x\right)=\frac{P\left(x\right)}{Q\left(x\right)}=\frac{{a}_{p}{x}^{p}+{a}_{p-1}{x}^{p-1}+...+{a}_{1}x+{a}_{0}}{{b}_{q}{x}^{q}+{b}_{q-1}{x}^{q-1}+...+{b}_{1}x+{b}_{0}},Q\left(x\right)\ne 0$

## Solving an applied problem involving a rational function

A large mixing tank currently contains 100 gallons of water into which 5 pounds of sugar have been mixed. A tap will open pouring 10 gallons per minute of water into the tank at the same time sugar is poured into the tank at a rate of 1 pound per minute. Find the concentration (pounds per gallon) of sugar in the tank after 12 minutes. Is that a greater concentration than at the beginning?

Let $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ be the number of minutes since the tap opened. Since the water increases at 10 gallons per minute, and the sugar increases at 1 pound per minute, these are constant rates of change. This tells us the amount of water in the tank is changing linearly, as is the amount of sugar in the tank. We can write an equation independently for each:

The concentration, $\text{\hspace{0.17em}}C,\text{\hspace{0.17em}}$ will be the ratio of pounds of sugar to gallons of water

$C\left(t\right)=\frac{5+t}{100+10t}$

The concentration after 12 minutes is given by evaluating $\text{\hspace{0.17em}}C\left(t\right)\text{\hspace{0.17em}}$ at

$\begin{array}{ccc}\hfill C\left(12\right)& =& \frac{5+12}{100+10\left(12\right)}\hfill \\ & =& \frac{17}{220}\hfill \end{array}$

This means the concentration is 17 pounds of sugar to 220 gallons of water.

At the beginning, the concentration is

$\begin{array}{ccc}\hfill C\left(0\right)& =& \frac{5+0}{100+10\left(0\right)}\hfill \\ & =& \frac{1}{20}\hfill \end{array}$

Since $\text{\hspace{0.17em}}\frac{17}{220}\approx 0.08>\frac{1}{20}=0.05,\text{\hspace{0.17em}}$ the concentration is greater after 12 minutes than at the beginning.

There are 1,200 freshmen and 1,500 sophomores at a prep rally at noon. After 12 p.m., 20 freshmen arrive at the rally every five minutes while 15 sophomores leave the rally. Find the ratio of freshmen to sophomores at 1 p.m.

$\frac{12}{11}$

## Finding the domains of rational functions

A vertical asymptote    represents a value at which a rational function is undefined, so that value is not in the domain of the function. A reciprocal function cannot have values in its domain that cause the denominator to equal zero. In general, to find the domain of a rational function, we need to determine which inputs would cause division by zero.

## Domain of a rational function

The domain of a rational function includes all real numbers except those that cause the denominator to equal zero.

Given a rational function, find the domain.

1. Set the denominator equal to zero.
2. Solve to find the x -values that cause the denominator to equal zero.
3. The domain is all real numbers except those found in Step 2.

## Finding the domain of a rational function

Find the domain of $\text{\hspace{0.17em}}f\left(x\right)=\frac{x+3}{{x}^{2}-9}.$

Begin by setting the denominator equal to zero and solving.

$\begin{array}{ccc}\hfill {x}^{2}-9& =& 0\hfill \\ \hfill {x}^{2}& =& 9\hfill \\ \hfill x& =& ±3\hfill \end{array}$

The denominator is equal to zero when $\text{\hspace{0.17em}}x=±3.\text{\hspace{0.17em}}$ The domain of the function is all real numbers except $\text{\hspace{0.17em}}x=±3.$

Find the domain of $\text{\hspace{0.17em}}f\left(x\right)=\frac{4x}{5\left(x-1\right)\left(x-5\right)}.$

The domain is all real numbers except $\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=5.$

## Identifying vertical asymptotes of rational functions

By looking at the graph of a rational function, we can investigate its local behavior and easily see whether there are asymptotes. We may even be able to approximate their location. Even without the graph, however, we can still determine whether a given rational function has any asymptotes, and calculate their location.

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
More example of algebra and trigo
What is Indices
If one side only of a triangle is given is it possible to solve for the unkown two sides?
cool
Rubben
kya
Khushnama
please I need help in maths
Okey tell me, what's your problem is?
Navin