MATH 1324 Section 10.3: Rational Functions

Introduction to Rational Functions

Definition: A rational function is expressed as ( R(x) = \frac{P(x)}{Q(x)} ), where both ( P(x) ) and ( Q(x) ) are polynomials, and ( Q(x) \neq 0 ).
Domain: The domain of a rational function includes all real numbers except those that make the denominator zero.

Vertical Asymptotes:
- Occur at the values of ( x ) which make ( Q(x) = 0 ) (assuming ( P(x) \neq 0 ) at those points).
- Indicate where the function heads toward infinity.
Horizontal Asymptotes:
- Determined by the degrees of ( P(x) ) and ( Q(x) ).
- Rules:
  - If degree of ( P(x) < Q(x) ), horizontal asymptote is ( y = 0 ).
  - If degree of ( P(x) = Q(x) ), horizontal asymptote is ( y = \frac{a}{b} ), where ( a ) and ( b ) are leading coefficients.
  - If degree of ( P(x) > Q(x) ), no horizontal asymptote (oblique asymptote may exist).
Holes:
- Occur when ( P(x) ) and ( Q(x) ) have common factors. The hole exists at the common factor location.

Steps:
1. Identify and plot the vertical and horizontal asymptotes.
2. Identify and plot any holes by finding common factors between numerator and denominator.
3. Plot additional points to determine the shape of the graph.
4. Connect the points, considering the behavior near the asymptotes and holes.

Example 1: Finding asymptotes for ( R(x) = \frac{x^2 - 4}{x^2 - 1} ).
- Vertical asymptotes at ( x = 1 ) and ( x = -1 ) (denominator zero).
- Horizontal asymptote at ( y = 1 ) (degrees equal).
- No holes since no common factors.
Problem-solving techniques:
- Identify domain restrictions and asymptotes first.
- Look for and cancel common factors to find holes.
- Use test points to sketch the graph.

Rational functions have unique characteristics determined by their numerator and denominator.
Understanding how to find asymptotes and holes is crucial for graphing.
Practice with different types of rational functions to become proficient in analyzing and graphing.