E4, E5

Rational Functions: Domain, Intercepts, Asymptotes, and Holes

Introduction to Rational Functions

A rational function is a function of the form $f(x) = \frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials and $q(x) \neq 0$. The study of rational functions involves analyzing their domains, intercepts, asymptotes, and holes, both algebraically and graphically.

Domain and Intercepts of Rational Functions

Domain

• Domain: The set of all real numbers except those that make the denominator zero.

• To find the domain, solve $q(x) = 0$ and exclude those $x$-values from the domain.

• Example: For $f(x) = \frac{x-2}{x^2-4}$, set $x^2-4=0 \implies x=2$ or $x=-2$. So, the domain is $x \in \mathbb{R}, x \neq 2, x \neq -2$.

Intercepts

• x-intercept: Set $f(x) = 0$ (i.e., set the numerator equal to zero and solve for $x$).

• y-intercept: Set $x = 0$ and solve for $f(0)$.

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

  • x-intercept: $x-2=0 \implies x=2$ (but $x=2$ is not in the domain, so no x-intercept).

  • y-intercept: $f(0) = \frac{0-2}{0^2-4} = \frac{-2}{-4} = \frac{1}{2}$.

Vertical, Horizontal, and Slant Asymptotes

Vertical Asymptotes

• Occur at values of $x$ where the denominator is zero and the numerator is not zero.

• To find, solve $q(x) = 0$ and check that $p(x) \neq 0$ at those points.

• Example: For $f(x) = \frac{x+1}{x^2-4}$, vertical asymptotes at $x=2$ and $x=-2$.

Horizontal Asymptotes

• Describe the end behavior of the function as $x \to \infty$ or $x \to -\infty$.

• Rules:

  • If degree of numerator < degree of denominator, horizontal asymptote at $y=0$.

  • If degrees are equal, horizontal asymptote at $y=\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$.

  • If degree of numerator > degree of denominator, no horizontal asymptote (may have a slant asymptote).

• Example: For $f(x) = \frac{2x^2+1}{x^2-4}$, degrees are equal, so horizontal asymptote at $y=\frac{2}{1}=2$.

Slant (Oblique) Asymptotes

• Occur when the degree of the numerator is exactly one more than the degree of the denominator.

• To find, perform polynomial long division of numerator by denominator. The quotient (without the remainder) is the equation of the slant asymptote.

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

Holes in the Graph of Rational Functions

• A hole occurs at $x=a$ if both the numerator and denominator have a common factor $(x-a)$.

• To find holes:

  1. Factor numerator and denominator.

  2. Identify and cancel common factors.

  3. The $x$-value where the common factor is zero is the location of the hole.

  4. To find the $y$-value, substitute $x=a$ into the reduced function.

• Example: $f(x) = \frac{(x-2)(x+1)}{(x-2)(x-3)}$ has a hole at $x=2$.

Summary Table: Asymptotes of Rational Functions

Type of Asymptote	How to Find	Equation
Vertical	Set denominator = 0 (after canceling common factors)	$x = a$
Horizontal	Compare degrees of numerator and denominator	$y = 0$ or $y = \frac{a_n}{b_m}$
Slant (Oblique)	Degree of numerator is one more than denominator; use long division	$y = mx + b$

Worked Examples

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

• Domain: $x \neq 2, x \neq -2$

• Vertical Asymptotes: $x=2, x=-2$

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

• x-intercept: None (since $x=2$ not in domain)

• y-intercept: $\frac{1}{2}$

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

• Domain: $x \neq 2, x \neq -2$

• Vertical Asymptotes: $x=2, x=-2$

• Horizontal Asymptote: $y=0$

• x-intercept: $x=-1$

• y-intercept: $-\frac{1}{4}$

Example 3: $f(x) = \frac{x^2-1}{x^2-4}$

• Domain: $x \neq 2, x \neq -2$

• Vertical Asymptotes: $x=2, x=-2$

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

• Holes: None (no common factors)

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

• Domain: $x \neq 1, x \neq -1$

• Vertical Asymptotes: $x=1, x=-1$

• Horizontal Asymptote: $y=1$

• Holes: None

Example 5: $f(x) = \frac{2x^2+2}{x^2-4}$

• Domain: $x \neq 2, x \neq -2$

• Vertical Asymptotes: $x=2, x=-2$

• Horizontal Asymptote: $y=2$

Example 6: $f(x) = \frac{x^2-1}{x-2}$

• Domain: $x \neq 2$

• Vertical Asymptote: $x=2$

• Slant Asymptote: $y=x+2$ (from long division)

Practice Problems

• Find the equations of the vertical, horizontal, and slant asymptotes for given rational functions.

• Determine the domain, intercepts, and holes for each function.

• Sketch the graph, indicating all asymptotes and holes.

Summary

• To analyze a rational function, always start by factoring numerator and denominator.

• Identify domain restrictions, intercepts, asymptotes, and holes systematically.

• Use algebraic and graphical methods to confirm your results.