# 5.6 Rational functions

 Page 1 / 16
In this section, you will:
• Use arrow notation.
• Solve applied problems involving rational functions.
• Find the domains of rational functions.
• Identify vertical asymptotes.
• Identify horizontal asymptotes.
• Graph rational functions.

Suppose we know that the cost of making a product is dependent on the number of items, $\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}$ produced. This is given by the equation $\text{\hspace{0.17em}}C\left(x\right)=15,000x-0.1{x}^{2}+1000.\text{\hspace{0.17em}}$ If we want to know the average cost for producing $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ items, we would divide the cost function by the number of items, $\text{\hspace{0.17em}}x.$

The average cost function, which yields the average cost per item for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ items produced, is

$f\left(x\right)=\frac{15,000x-0.1{x}^{2}+1000}{x}$

Many other application problems require finding an average value in a similar way, giving us variables in the denominator. Written without a variable in the denominator, this function will contain a negative integer power.

In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.

## Using arrow notation

We have seen the graphs of the basic reciprocal function and the squared reciprocal function from our study of toolkit functions. Examine these graphs, as shown in [link] , and notice some of their features.

Several things are apparent if we examine the graph of $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x}.$

1. On the left branch of the graph, the curve approaches the x -axis
2. As the graph approaches $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ from the left, the curve drops, but as we approach zero from the right, the curve rises.
3. Finally, on the right branch of the graph, the curves approaches the x- axis

To summarize, we use arrow notation    to show that $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ is approaching a particular value. See [link] .

Symbol Meaning
$x\to {a}^{-}$ $x\text{\hspace{0.17em}}$ approaches $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ from the left ( $x but close to $\text{\hspace{0.17em}}a$ )
$x\to {a}^{+}$ $x\text{\hspace{0.17em}}$ approaches $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ from the right ( $x>a\text{\hspace{0.17em}}$ but close to $\text{\hspace{0.17em}}a$ )
$x\to \infty$ $x\text{\hspace{0.17em}}$ approaches infinity ( $x\text{\hspace{0.17em}}$ increases without bound)
$x\to -\infty$ $x\text{\hspace{0.17em}}$ approaches negative infinity ( $x\text{\hspace{0.17em}}$ decreases without bound)
$f\left(x\right)\to \infty$ the output approaches infinity (the output increases without bound)
$f\left(x\right)\to -\infty$ the output approaches negative infinity (the output decreases without bound)
$f\left(x\right)\to a$ the output approaches $\text{\hspace{0.17em}}a$

## Local behavior of $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x}$

Let’s begin by looking at the reciprocal function, $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x}.\text{\hspace{0.17em}}$ We cannot divide by zero, which means the function is undefined at $\text{\hspace{0.17em}}x=0;\text{\hspace{0.17em}}$ so zero is not in the domain . As the input values approach zero from the left side (becoming very small, negative values), the function values decrease without bound (in other words, they approach negative infinity). We can see this behavior in [link] .

 $x$ –0.1 –0.01 –0.001 –0.0001 $f\left(x\right)=\frac{1}{x}$ –10 –100 –1000 –10,000

We write in arrow notation

As the input values approach zero from the right side (becoming very small, positive values), the function values increase without bound (approaching infinity). We can see this behavior in [link] .

 $x$ 0.1 0.01 0.001 0.0001 $f\left(x\right)=\frac{1}{x}$ 10 100 1000 10,000

We write in arrow notation

This behavior creates a vertical asymptote , which is a vertical line that the graph approaches but never crosses. In this case, the graph is approaching the vertical line $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ as the input becomes close to zero. See [link] .

show that the set of all natural number form semi group under the composition of addition
what is the meaning
Dominic
explain and give four Example hyperbolic function
_3_2_1
felecia
⅗ ⅔½
felecia
_½+⅔-¾
felecia
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
2+2x=
felecia
×/×+9+6/1
Debbie
Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22
Naagmenkoma
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Abdullahi
hi mam
Mark
find the value of 2x=32
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
16
Makan
x=16
Makan
use the y -intercept and slope to sketch the graph of the equation y=6x
how do we prove the quadratic formular
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak