# 5.6 Rational functions  (Page 4/16)

 Page 4 / 16

## Vertical asymptotes

The vertical asymptotes of a rational function may be found by examining the factors of the denominator that are not common to the factors in the numerator. Vertical asymptotes occur at the zeros of such factors.

Given a rational function, identify any vertical asymptotes of its graph.

1. Factor the numerator and denominator.
2. Note any restrictions in the domain of the function.
3. Reduce the expression by canceling common factors in the numerator and the denominator.
4. Note any values that cause the denominator to be zero in this simplified version. These are where the vertical asymptotes occur.
5. Note any restrictions in the domain where asymptotes do not occur. These are removable discontinuities, or “holes.”

## Identifying vertical asymptotes

Find the vertical asymptotes of the graph of $\text{\hspace{0.17em}}k\left(x\right)=\frac{5+2{x}^{2}}{2-x-{x}^{2}}.$

First, factor the numerator and denominator.

$\begin{array}{ccc}\hfill k\left(x\right)& =& \frac{5+2{x}^{2}}{2-x-{x}^{2}}\hfill \\ & =& \frac{5+2{x}^{2}}{\left(2+x\right)\left(1-x\right)}\hfill \end{array}$

To find the vertical asymptotes, we determine where this function will be undefined by setting the denominator equal to zero:

$\begin{array}{ccc}\hfill \left(2+x\right)\left(1-x\right)& =& 0\hfill \\ \hfill x& =& -2,1\hfill \end{array}$

Neither $\text{\hspace{0.17em}}x=–2\text{\hspace{0.17em}}$ nor $\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ are zeros of the numerator, so the two values indicate two vertical asymptotes. The graph in [link] confirms the location of the two vertical asymptotes.

## Removable discontinuities

Occasionally, a graph will contain a hole: a single point where the graph is not defined, indicated by an open circle. We call such a hole a removable discontinuity    .

For example, the function $\text{\hspace{0.17em}}f\left(x\right)=\frac{{x}^{2}-1}{{x}^{2}-2x-3}\text{\hspace{0.17em}}$ may be re-written by factoring the numerator and the denominator.

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

Notice that $\text{\hspace{0.17em}}x+1\text{\hspace{0.17em}}$ is a common factor to the numerator and the denominator. The zero of this factor, $\text{\hspace{0.17em}}x=-1,\text{\hspace{0.17em}}$ is the location of the removable discontinuity. Notice also that $\text{\hspace{0.17em}}x–3\text{\hspace{0.17em}}$ is not a factor in both the numerator and denominator. The zero of this factor, $\text{\hspace{0.17em}}x=3,\text{\hspace{0.17em}}$ is the vertical asymptote. See [link] . [Note that removable discontinuities may not be visible when we use a graphing calculator, depending upon the window selected.]

## Removable discontinuities of rational functions

A removable discontinuity    occurs in the graph of a rational function at $\text{\hspace{0.17em}}x=a\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is a zero for a factor in the denominator that is common with a factor in the numerator. We factor the numerator and denominator and check for common factors. If we find any, we set the common factor equal to 0 and solve. This is the location of the removable discontinuity. This is true if the multiplicity of this factor is greater than or equal to that in the denominator. If the multiplicity of this factor is greater in the denominator, then there is still an asymptote at that value.

## Identifying vertical asymptotes and removable discontinuities for a graph

Find the vertical asymptotes and removable discontinuities of the graph of $\text{\hspace{0.17em}}k\left(x\right)=\frac{x-2}{{x}^{2}-4}.$

Factor the numerator and the denominator.

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

Notice that there is a common factor in the numerator and the denominator, $\text{\hspace{0.17em}}x–2.\text{\hspace{0.17em}}$ The zero for this factor is $\text{\hspace{0.17em}}x=2.\text{\hspace{0.17em}}$ This is the location of the removable discontinuity.

Notice that there is a factor in the denominator that is not in the numerator, $\text{\hspace{0.17em}}x+2.\text{\hspace{0.17em}}$ The zero for this factor is $\text{\hspace{0.17em}}x=-2.\text{\hspace{0.17em}}$ The vertical asymptote is $\text{\hspace{0.17em}}x=-2.\text{\hspace{0.17em}}$ See [link] .

The graph of this function will have the vertical asymptote at $\text{\hspace{0.17em}}x=-2,\text{\hspace{0.17em}}$ but at $\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ the graph will have a hole.

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
Y
master
master
Soo sorry (5±Root11* i)/3
master
Mukhtar
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
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
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
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
Jeffrey
Jeffrey
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
Miranda
Jeffrey