# 3.7 Rational functions  (Page 10/16)

 Page 10 / 16

## Key equations

 Rational Function

## Key concepts

• We can use arrow notation to describe local behavior and end behavior of the toolkit functions $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{{x}^{2}}.\text{\hspace{0.17em}}$ See [link] .
• A function that levels off at a horizontal value has a horizontal asymptote. A function can have more than one vertical asymptote. See [link] .
• Application problems involving rates and concentrations often involve rational functions. See [link] .
• The domain of a rational function includes all real numbers except those that cause the denominator to equal zero. See [link] .
• The vertical asymptotes of a rational function will occur where the denominator of the function is equal to zero and the numerator is not zero. See [link] .
• A removable discontinuity might occur in the graph of a rational function if an input causes both numerator and denominator to be zero. See [link] .
• A rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions. See [link] , [link] , [link] , and [link] .
• Graph rational functions by finding the intercepts, behavior at the intercepts and asymptotes, and end behavior. See [link] .
• If a rational function has x -intercepts at $\text{\hspace{0.17em}}x={x}_{1},{x}_{2},\dots ,{x}_{n},\text{\hspace{0.17em}}$ vertical asymptotes at $\text{\hspace{0.17em}}x={v}_{1},{v}_{2},\dots ,{v}_{m},\text{\hspace{0.17em}}$ and no then the function can be written in the form
$\begin{array}{l}\begin{array}{l}\hfill \\ f\left(x\right)=a\frac{{\left(x-{x}_{1}\right)}^{{p}_{1}}{\left(x-{x}_{2}\right)}^{{p}_{2}}\cdots {\left(x-{x}_{n}\right)}^{{p}_{n}}}{{\left(x-{v}_{1}\right)}^{{q}_{1}}{\left(x-{v}_{2}\right)}^{{q}_{2}}\cdots {\left(x-{v}_{m}\right)}^{{q}_{n}}}\hfill \end{array}\hfill \end{array}$

## Verbal

What is the fundamental difference in the algebraic representation of a polynomial function and a rational function?

The rational function will be represented by a quotient of polynomial functions.

What is the fundamental difference in the graphs of polynomial functions and rational functions?

If the graph of a rational function has a removable discontinuity, what must be true of the functional rule?

The numerator and denominator must have a common factor.

Can a graph of a rational function have no vertical asymptote? If so, how?

Can a graph of a rational function have no x -intercepts? If so, how?

Yes. The numerator of the formula of the functions would have only complex roots and/or factors common to both the numerator and denominator.

## Algebraic

For the following exercises, find the domain of the rational functions.

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

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

$f\left(x\right)=\frac{{x}^{2}+4}{{x}^{2}-2x-8}$

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

For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions.

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

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

V.A. at $\text{\hspace{0.17em}}x=–\frac{2}{5};\text{\hspace{0.17em}}$ H.A. at $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ Domain is all reals $\text{\hspace{0.17em}}x\ne –\frac{2}{5}$

$f\left(x\right)=\frac{x}{{x}^{2}-9}$

$f\left(x\right)=\frac{x}{{x}^{2}+5x-36}$

V.A. at H.A. at $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ Domain is all reals

$f\left(x\right)=\frac{3+x}{{x}^{3}-27}$

$f\left(x\right)=\frac{3x-4}{{x}^{3}-16x}$

V.A. at H.A. at $\text{\hspace{0.17em}}y=0;$ Domain is all reals

$f\left(x\right)=\frac{{x}^{2}-1}{{x}^{3}+9{x}^{2}+14x}$

$f\left(x\right)=\frac{x+5}{{x}^{2}-25}$

V.A. at $\text{\hspace{0.17em}}x=-5;\text{\hspace{0.17em}}$ H.A. at $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ Domain is all reals $\text{\hspace{0.17em}}x\ne 5,-5$

$f\left(x\right)=\frac{x-4}{x-6}$

$f\left(x\right)=\frac{4-2x}{3x-1}$

V.A. at $\text{\hspace{0.17em}}x=\frac{1}{3};\text{\hspace{0.17em}}$ H.A. at $\text{\hspace{0.17em}}y=-\frac{2}{3};\text{\hspace{0.17em}}$ Domain is all reals $\text{\hspace{0.17em}}x\ne \frac{1}{3}.$

For the following exercises, find the x - and y -intercepts for the functions.

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

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

none

$f\left(x\right)=\frac{{x}^{2}+8x+7}{{x}^{2}+11x+30}$

$f\left(x\right)=\frac{{x}^{2}+x+6}{{x}^{2}-10x+24}$

$f\left(x\right)=\frac{94-2{x}^{2}}{3{x}^{2}-12}$

For the following exercises, describe the local and end behavior of the functions.

what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
can I see the picture
How would you find if a radical function is one to one?