# 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.