# 5.6 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.

By the definition, is such that 0!=1.why?
(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
hatdog
Mark
how we can draw three triangles of distinctly different shapes. All the angles will be cutt off each triangle and placed side by side with vertices touching
bsc F. y algebra and trigonometry pepper 2
given that x= 3/5 find sin 3x
4
DB
remove any signs and collect terms of -2(8a-3b-c)
-16a+6b+2c
Will
Joeval
(x2-2x+8)-4(x2-3x+5)
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
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master
master
Soo sorry (5±Root11* i)/3
master
Mukhtar
2x²-6x+1=0
Ife
explain and give four example of hyperbolic function
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
y2=4ax= y=4ax/2. y=2ax
akash
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
A banana.
Yaona
a function
Daniel
a function
emmanuel
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda