# 3.7 Rational functions

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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] .

Arrow notation
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] .

what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich