E4-E5

Rational Functions: Domain, Intercepts, Asymptotes, and Holes

Introduction to Rational Functions

A rational function is any function that can be written as the ratio of two polynomials, that is, in the form:

$f(x) = \frac{p(x)}{q(x)}$

• Domain: All real numbers except those that make the denominator zero.

• Intercepts: Points where the graph crosses the axes.

• Asymptotes: Lines that the graph approaches but never touches.

• Holes: Points where the function is not defined due to common factors in numerator and denominator.

Domain of Rational Functions

Finding the Domain

The domain of a rational function consists of all real numbers except those that make the denominator zero.

• Set the denominator equal to zero and solve for x.

• Exclude these values from the domain.

Example: For $f(x) = \frac{x+1}{x^2-9}$, set $x^2-9=0$ so $x=3$ or $x=-3$. Thus, the domain is all real numbers except $x=3$ and $x=-3$.

Intercepts of Rational Functions

Finding x- and y-Intercepts

• x-intercept: Set the numerator equal to zero and solve for x (as long as the denominator is not zero at that value).

• y-intercept: Substitute $x=0$ into the function and simplify.

Example: For $f(x) = \frac{x+1}{x^2-9}$:

• x-intercept: $x+1=0 \implies x=-1$ (as long as $x=-1$ does not make denominator zero).

• y-intercept: $f(0) = \frac{0+1}{0^2-9} = \frac{1}{-9}$.

Vertical Asymptotes

Definition and Identification

A vertical asymptote occurs at values of x that make the denominator zero, provided these values do not also make the numerator zero (which would indicate a hole instead).

• Set the denominator equal to zero and solve for x.

• If the factor does not cancel with the numerator, it is a vertical asymptote.

Example: For $f(x) = \frac{x+1}{x^2-9}$, vertical asymptotes at $x=3$ and $x=-3$.

Holes in the Graph

Definition and Identification

A hole in the graph occurs at values of x where both the numerator and denominator are zero (i.e., they share a common factor).

• Factor both numerator and denominator.

• If a factor cancels, the graph has a hole at that x-value.

Example: For $f(x) = \frac{(x-2)(x+1)}{(x-2)(x+3)}$, $x=2$ is a hole because $(x-2)$ cancels.

Horizontal and Slant (Oblique) Asymptotes

Horizontal Asymptotes

Horizontal asymptotes describe the end behavior of a rational function as $x$ approaches infinity or negative infinity. The rules depend on the degrees of the numerator ($n$) and denominator ($m$):

Degree of Numerator ($n$)	Degree of Denominator ($m$)	Horizontal Asymptote
$n < m$		$y = 0$
$n = m$		$y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$
$n > m$		No horizontal asymptote (may have a slant asymptote)

Slant (Oblique) Asymptotes

If the degree of the numerator is exactly one more than the degree of the denominator ($n = m + 1$), the function has a slant (oblique) asymptote. Find it by dividing the numerator by the denominator using polynomial long division.

Example: For $f(x) = \frac{x^2+1}{x+1}$, divide $x^2+1$ by $x+1$ to get the slant asymptote $y = x - 1$.

Summary Table: Asymptote Rules

Type	How to Find
Vertical Asymptote	Set denominator = 0 (after canceling common factors)
Horizontal Asymptote	Compare degrees of numerator and denominator
Slant Asymptote	If numerator degree is one more than denominator, use long division

Worked Examples

Example 1

For $f(x) = \frac{x+1}{x^2-9}$:

• Domain: $x \neq 3, -3$

• x-intercept: $x = -1$

• y-intercept: $y = -\frac{1}{9}$

• Vertical asymptotes: $x = 3, -3$

• Horizontal asymptote: $y = 0$ (degree numerator < degree denominator)

Example 2

For $f(x) = \frac{x+1}{x^2-2x-3}$:

• Factor denominator: $x^2-2x-3 = (x-3)(x+1)$

• Domain: $x \neq 3, -1$

• Vertical asymptote: $x = 3$

• Hole: $x = -1$ (since $x+1$ cancels)

Example 3

For $f(x) = \frac{x^2-1}{x^2-4}$:

• Factor numerator and denominator: $x^2-1 = (x-1)(x+1)$, $x^2-4 = (x-2)(x+2)$

• Domain: $x \neq 2, -2$

• Vertical asymptotes: $x = 2, -2$

• Horizontal asymptote: $y = 1$ (degrees equal, leading coefficients both 1)

Practice Problems

1. Find the equations of the vertical and horizontal asymptotes for $f(x) = \frac{x+2}{x-3}$.

2. Find the slant asymptote for $f(x) = \frac{x^2-2}{x-1}$.

3. For $f(x) = \frac{x^2-1}{x^2-4}$, find the domain, intercepts, asymptotes, and sketch the graph.

Key Takeaways

• Always factor numerator and denominator to identify holes and asymptotes.

• Use the degree of numerator and denominator to determine horizontal or slant asymptotes.

• Check for holes before identifying vertical asymptotes.