2.6 rational functions 

Definition of Rational Functions

• Rational functions are defined as ratios of polynomial expressions.
• Examples:
  • $(\frac{x^2 + 7x - 12}{x^3 - 63})$
• The key characteristic is that they create fractions based on polynomial expressions.

Characteristics of Rational Functions

• Types of Expressions Involved:
  • Radicals: Not included in rational functions.
  • Exponents: Non-fractional and non-negative exponents only.
  • Imaginary Numbers: Not part of the study of rational functions within this context.

Domain of Rational Functions

• The domain consists of all real numbers except those that make the denominator zero.
• Determining the domain requires:
  • Identifying values of x that cause the denominator to be zero.
• Example Analysis:
  • For $\frac{x^2 + 7x - 12}{x^3 - 63}$: the domain is all real numbers except where $x^3 - 63 = 0$.
• General approach: 'Set the denominator (denoted as $q(x)$) equal to zero to find exclusions from the domain.

Finding the Domain: Process Steps

1. Identify the Denominator: Focus solely on the denominator to establish exclusions.
2. Set the Denominator to Zero: For $F(x) = 3x^2$, exclude values where $q(x) = 0$.
   • Method of Solving Quadratic Equations: Use factoring whenever possible.
3. Recognize Factorization: Identify common factors.
4. Applies to All Rational Functions: The potential exclusions must be noted for understanding behaviors like asymptotes and holes.

Identifying Asymptotes and Holes

• Vertical Asymptotes: occur when the denominator has a factor that is not removable.
• Holes: occur if factors can be canceled out (removable discontinuities).
• Graphical Representation:
  • Vertical asymptotes correspond to restrictions in the domain.
  • Holes represent removable discontinuities in the graph.

Understanding Behavior Near Excluded Values

1. Determine if the excluded value leads to a vertical asymptote or a hole.
2. Vertical asymptotes cause the function to diverge towards infinity or negative infinity.
3. Holes represent points where the function simplifies to an undefined value but can be evaluated through other means.

Practical Examples

• Nonremovable Discontinuity:
  • Example: $f(x) = \frac{x - 3}{(x - 2)(x - 3)}$
  • Here, $x - 3$ is a removable factor, while $x - 2$ is non-removable, revealing a vertical asymptote at $x = 2$ and a hole at $x = 3$.
• Evaluating the Graph: - Vertical asymptote at $x = 2$ will guide how the function behaves at that point, while the hole at $3$ will also be noted.

Behavior near Vertical Asymptotes

• To investigate what happens around vertical asymptotes, set up a table considering points close to the asymptote.
• Example Calculations:
  • As $x$ approaches an asymptote from either side, observe whether the functional values approach infinity or negative infinity.
  • Limit Notation: To convey long-term behaviors:
  • $ext{If } x o a, F(x) o ext{infinity/negative infinity}$

Final Remarks and Assignments

• The study session's homework includes saying identifying domain correctly, recognizing asymptotes and holes, and behaviors near excluded values.
• Problems may require investigating the graphical implications of rational functions to understand the behavior adequately.