**2.6 Rational Functions and Asymptotes**

A rational function can be written in the form:

f(x) = N(x)/D(

*x*)

where

*N*(

*x*) is the numerator and

*D*(

*x*) is the denominator and both are polynomials but

*D*(

*x*) is not the zero polynomial (because you cannot divide by zero)

**I. Asymptotes of a Rational Function:**

Let "f" be the rational function

*f*(

*x*) where

*f*(

*x*) =

*N*(

*x*)/

*D*(

*x*) where

N(x) = a

_{n}x

^{n}+ a

_{n-1}x

^{n-1}+ ... + a

_{1}x + a

_{0}and

D(x) = b

_{m}x

^{m}+ b

_{m-1}x

^{m-1}+ ... + b

_{1}x + b

_{0}

**Where N(x) and D(x) have no common factors.**

**1.**The graph of "

*f*" has vertical asymptotes at the zeros of

*D*(

*x*).

**2.**The graph of "

*f*" has at most one horizontal asymptote determined by comparing the degrees of

*N*(

*x*) and

*D*(

*x*).

**a**. If

*is less than*

**n****, the line**

*m**y*= 0 (

*x*-axis) is a horizontal asymptote.

**b.**If

**, the line**

*n*=*m**y*= (

*a*

_{n})/(

*b*

_{m}) is a horizontal asymptote.

**c.**If

**is greater than**

*n***, the graph of "**

*m**f*" has no horizontal asymptote.

**Examples:**

**a)**f(x) = (5x

^{2}+ 3)/(-6x

^{3}+ 2x + 4)

*n*= 2 and

*m*= 3 which means that n is less than m so the horizontal asymptote is y = 0

**b)**f(x) = (2x

^{3}+ 2x

^{2})/(3x

^{3}+ 4x)

*n*= 3 and

*m*= 3 so the line y = 2/3 is a horizontal asymptote.

**c)**f (x) = (3x

^{4}+ 2x

^{2}+ 5)/(4x

^{3}+ 3x)

*n*= 4 and

*m*= 3 so

*n*is less than

*m*, therefore there is not a horizontal asymptote.

**II. Find the Domain and Asymptotes**of

*f*(

*x*) = 3/ ((

*x*- 2)

^{3})

1. Find the vertical asymptotes by taking D(x) and setting it equal to zero.

(x - 2)

^{3}= 0

x - 2 = 0

x = 2

therefore a vertical asymptote is the line x = 2

2. Find the horizontal asymptotes

*n*= 0 and

*m*= 3 so

*n*is less than

*m*, therefore the line

*y*= 0 is the horizontal asymptote.

3. Graph it using a graphing utility.

4. Domain (- ∞, 2) U (2, ∞)

5. Range (- ∞, 0) U (0, ∞)

6. check using a table:

{ (-3, -0.24), (-2, -.047), (-1, -.11), (0, -3.75), (1, -3), (2, error), (3, 3)}

**Homework:**Quiz (19, 20)

Pg. 187/ 7, 21, 23, 43*Pg. 195/ 13 – 19 odd, 27 – 31 odd, 35