Study Notes for Rational Functions and Asymptotes

The chapter is divided into two major topics:
- Polynomials
- Rational Functions

Definition: A rational function can be expressed as the ratio of two polynomials.
Characteristics:
- Unlike polynomials, rational functions have different characteristics when it comes to graphing.
- Essential to understand polynomials' behaviors and end behaviors since rational functions will deviate from this.

Domain: The set of values for which the function is defined.
For a rational function $\frac{p(x)}{q(x)}$ , where both $p(x)$ and $q(x)$ are polynomials, the domain is restricted to the values of $x$ that do not make the denominator zero.
Finding the Domain:
- Identify the values of $x$ that make the denominator equal to zero.
- Set Builder Notation: E.g., for $\frac{1}{x + 4}$ , the domain can be written as:
- ${ x | x \neq -4 }$
- Interval Notation: E.g., All $x$ except -4 can be expressed as:
- $(-\infty, -4) \cup (-4, \infty)$

Example 1: $\frac{1}{x + 4}$
- Domain: $x \neq -4$
Example 2: $\frac{1}{(x - 2)(x + 2)}$
- Domain: $x \neq 2, \; x \neq -2$
- Set Builder: ${x | x \neq 2 \; \land \; x \neq -2}$
Example 3: $\frac{1}{x^2 + 1}$
- Domain: All real numbers since $x^2 + 1$ is never zero.

Function Definition: The basic reciprocal function is expressed as $f(x) = \frac{1}{x}$ .
Asymptotes:
- Vertical Asymptote: Occurs when the denominator is zero.
- For $f(x) = \frac{1}{x}$ , vertical asymptote is at $x = 0$ (since $x$ cannot be zero).
- Horizontal Asymptote: Approaches $y = 0$ when $x$ approaches infinity, since the function never touches the x-axis.

Symmetric about the origin (odd function).
The graph of $f(x) = \frac{1}{x}$ has two distinct curves that approach the axes without touching them, the left curve in Quadrant III and the right curve in Quadrant I.

Rule: Vertical asymptotes occur at the zeros of the denominator.
- $\text{Equation: } x = k$ where $k$ is what makes the denominator zero.

Determined by comparing degrees:
- If Degree of numerator < Degree of denominator:
- Horizontal asymptote is at $y = 0$ .
- If Degree of numerator = Degree of denominator:
- Horizontal asymptote is at $y = \frac{a<em>n}{b</em>n}$ ,
  where $a<em>n$ and $b</em>n$ are leading coefficients of numerator and denominator.
- If Degree of numerator > Degree of denominator:
- An oblique (slant) asymptote exists: found by polynomial long division.

Example: Given $f(x) = \frac{x^2 + 2}{x + 1}$ ,
- Degree of numerator (2) > Degree of denominator (1); thus, find oblique asymptote via long division.
Another Example: $g(x) = \frac{2x^2 + 3}{x^2 - 4}$ ,
- Degrees are equal; hence, horizontal asymptote is $y = \frac{2}{1} = 2$ .

Example Function: $h(x) = \frac{3x^2 - 11x - 4}{x - 4}$
- Vertical Asymptote:
  - Zeros of $x - 4$ yield $x = 4$.
- Horizontal Asymptote:
  - Compare degrees: both are 2; thus, the asymptote is: $y = \frac{3}{1} = 3$ .
Example Function with Hole:
- If a function simplifies and has a common factor in numerator and denominator, it indicates a hole where that factor equals zero.
- For instance, $k(x) = \frac{(x - 2)(x + 3)}{(x - 2)}$ exhibits a hole at $x = 2$ and the behavior of the graph near that point must be addressed.

Asymptotes dictate the behavior of the function graph in relation to the axes.
Understanding these characteristics is critical for accurate graph representation.
Finding intercepts, and analyzing behavior around vertical and horizontal asymptotes are foundational skills for examining rational functions.
Practicing with a calculator is encouraged, but understanding the underlying mathematical processes remains essential.

Review definitions and understand the steps: finding domain, vertical and horizontal asymptotes, and graphing.
Practice problems provided in the textbook for additional reinforcement.
Prepare for the upcoming test by ensuring familiarity with all concepts covered in chapters, focusing on asymptotes and their significance in rational functions.