# 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.

how can are find the domain and range of a relations
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
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
i want to sure my answer of the exercise
what is the diameter of(x-2)²+(y-3)²=25
how to solve the Identity ?
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
what is a complex number used for?
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim