Rational Functions

1. Rational Functions (Definition)

 Definition of a Rational Function A rational function is a quotient of polynomials P(x)/Q(x)

Examples

1. (x2 + x - 1)/(3x3 + 1)

2. (x - 1)/(x2 +1)

3. x2/(x + 1)

Example

Find the domain of

(x2 + 1)/(x2 -1)

The domain of this rational function is the set of all real numbers that do not make the denominator zero.  We find

x2 -1 = 0

solving

x = 1,     or     x = -1

So that the domain is

{x | x is not 1 or -1}

2. Vertical Asymptotes

 Definition of a Vertical Asymptote A Vertical Asymptote of a rational function occurs where the denominator is 0.

Example:
Graph the vertical asymptotes of

y = f(x) = (x2 + 1)/(x2 -1)

Solution

From the last example, we see that there are vertical asymptotes at 1 and -1.

Since f(x) is positive a little to the left of -1, we say that

as

x  -->  -1-
("x goes to -1 from the left")

f(x)   -->

Similarly since f(x) is negative a little to the right of -1, we say that

as

x-> -1+         ("x goes to -1 from the right")

f(x) -> -

Since f(x) is negative a little to the left of 1,

as

x  -->  1-

f(x)   -->    -

Similarly since f(x) is positive a little to the right of 1,

as

x-> 1+

f(x) ->

3. Four Types of Vertical Asymptotes

Below are the four types of vertical asymptotes:

4. Horizontal Asymptotes

Example:

Consider the rational function

f(x) = (3x2 + x - 1)/(x2 - x - 2)

For the numerator, the term 3x2 dominates when x is large, while for the denominator, the term x2 dominates when x is large.  Hence

as

x  -->

3 is called the horizontal asymptote and we have the the left and right behavior of the graph is a horizontal line y = 3.

5. Oblique Asymptotes

Consider the function

f(x) = (x2 - 3x - 4)/(x + 3)

f(x) does not have a horizontal asymptote, since

x2 / x = x

is not a constant, but we see (on the calculator) that the left and right behavior of the curve is like a line.  Our goal is to find the equation of this line.

We use synthetic division to see that

(x2 - 3x - 4)/(x + 3) = x - 6 + 14/(x + 3)

For very large x

14/x + 3

is very small, hence

f(x) is approximately equal to

x - 6

on the far left and far right of the graph.

We call this line an Oblique Asymptote

To graph by have we see that there is a vertical asymptote at

x = -3

with behavior:

left down and right up

The graph has x-intercepts at 4 and -1, and a y intercept at -4/3.

Exercise:

Graph

(x3 + 8)/(x2 - 3x - 4)

6. Rational Functions With Common Factors

Consider the graph of

y = (x - 1)/(x - 1)

What is wrong with the picture?

When

f(x) = g(x)(x - r)/h(x)(x - r)

with neither g(r) nor h(r) zero, the graph will have a hole at x = r.  We call this hole a removable discontinuity.

Example:

Graph

f(x) = (x2 - 2)/(x2 - x - 2) = (x - 2)(x + 2)/(x - 2)(x + 1)

This graph will have a vertical asymptote at x -1 and a hole at (2,2)

We end our discussion with a list of steps for graphing rational functions.

Steps in graphing rational functions:

Step 1     Plug in x = 0 to find the y-intercept

Step 2      Factor the numerator and denominator.  Cancel any common factors remember to put in the appropriate holes if necessary.

Step 3     Set the numerator = 0 to find the x-intercepts

Step 4     Set the denominator = 0 to find the vertical asymptotes.  Then plug in nearby values to fine the left and right behavior of the vertical asymptotes.

Step  5      If the degree of the numerator = degree of the denominator then the graph has a horizontal asymptote.  To determine the value of the horizontal asymptote, divide the term highest power of the numerator by the term of highest power of the denominator.

If the degree of the numerator = degree of the denominator + 1 then use polynomial or synthetic division to determine the equation of the oblique asymptote.

Step 6      Graph it!