# 3.7 Rational functions  (Page 4/16)

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## 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.

## 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.

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

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] .

## 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.

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
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