3.5: Rational Functions

1. Rational Functions (Definition)

Definition: Rational Function

A rational function is a quotient of polynomials \(\dfrac{P(x)}{Q(x)}\).

Example 1

\[\dfrac{(x^2 + x - 1)}{(3x^3+ 1)},\]

\[\dfrac{(x - 1)}{(x^2 +1)}, \text{ and}\]

\[\dfrac{x^2}{(x + 1)}\]

are all Rational Functions

Example 2

Find the domain of 

\[\dfrac{(x^2 + 1)}{(x^2 -1)}.\]

Solution

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

\[x^2 -1 = 0\]

solving

\[x = 1, \;\;\; \text{or} \;\;\; x = -1.\]

So that the domain is

\[\{x | x \text{ is not }1 \text{ or } -1\}.\]

2. Vertical Asymptotes

Definition: Vertical Asymptote

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

Example 3

Graph the vertical asymptotes of 

\[\dfrac{(x^2 + 1)}{(x^2 -1)}\]

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

\[x  \rightarrow  -1^{-} \text{ ("x goes to -1 from the left")},\]

\[f(x) \rightarrow \infty.\]

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

\[x\rightarrow -1^{+} \text{( "x goes to -1 from the right")}, \]

\[f(x) \rightarrow -\infty.\]

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

\[x \rightarrow1^{-},\]

\[f(x) \rightarrow -\infty.\]

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

\[x\rightarrow1^{+},\]

\[f(x) \rightarrow\infty.\]

Four Types of Vertical Asymptotes

Below are the four types of vertical asymptotes:

                       

3. Horizontal Asymptotes

Example 4

Consider the rational function

\[f(x) = \dfrac{(3x^2 + x - 1)}{(x^2 - x - 2)}.\]

For the numerator, the term \(3x^2\) dominates when \(x\) is large, while for the denominator, the term \(x^2\) dominates when \(x\) is large.  Hence as

\[x \rightarrow \infty,\]

\[f(x)\rightarrow\dfrac{3x^2}{x^2}=3.\]

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

4. Oblique Asymptotes

Consider the function

\[f(x) = \dfrac{(x^2 - 3x - 4)}{(x + 3)}\]

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

\[\dfrac{x^2}{x}= x \]

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

\[\dfrac{(x^2 - 3x - 4)}{(x + 3)} = x - 6 + \dfrac{14}{(x+3)}.\]

For very large \(x\),

\[\dfrac{14}{x} + 3\]

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, 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 \(-\frac{4}{3}\).  

Exercise

Graph

\[\dfrac{(x^3 + 8)}{(x^2 - 3x - 4)}\]

5. Rational Functions With Common Factors

Consider the graph of

\[y = \dfrac{x-1}{x-1}\]

What is wrong with the picture? When

\[f(x) = \dfrac{g(x)(x - r)}{h(x)(x - r)}\]

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

\[\begin{align} f(x) &= \dfrac{(x^2 - 2)}{(x^2 - x - 2)} = \dfrac{(x - 2)(x + 2)}{(x - 2)(x + 1)}.\end{align}\]

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!

Larry Green (Lake Tahoe Community College)

• Integrated by Justin Marshall.