Improper rational function

Definition

An improper rational function is a type of rational function where the degree of the numerator is greater than or equal to the degree of the denominator. This characteristic leads to specific behaviors in graphing, such as vertical asymptotes and horizontal asymptotes, which are key in understanding the overall shape and limits of the function as the input approaches certain values.

5 Must Know Facts For Your Next Test

1. Improper rational functions can exhibit behavior such as having vertical asymptotes when the denominator equals zero and horizontal asymptotes that indicate end behavior.
2. When dividing an improper rational function, polynomial long division can be used to express it as a sum of a polynomial and a proper rational function.
3. The degree of the numerator compared to the denominator helps determine whether there is no horizontal asymptote, a horizontal asymptote at zero, or a horizontal asymptote at a non-zero value.
4. For improper rational functions with degrees differing by one, there is typically an oblique asymptote rather than a horizontal one.
5. Understanding improper rational functions is crucial for analyzing limits, continuity, and behaviors of complex functions in calculus.

Review Questions

• What are the characteristics of improper rational functions compared to proper rational functions?
  • Improper rational functions differ from proper rational functions primarily in their degrees. In improper rational functions, the degree of the numerator is greater than or equal to that of the denominator, while in proper rational functions, the degree of the numerator is less than that of the denominator. This difference results in distinct graphing behaviors, such as having different types of asymptotes and potentially exhibiting more complex end behaviors as values approach infinity.
• How does polynomial long division help in analyzing improper rational functions?
  • Polynomial long division is used to simplify improper rational functions by breaking them down into a polynomial plus a proper rational function. This process allows for easier analysis of their end behavior and asymptotic characteristics. After performing long division, one can identify any oblique or horizontal asymptotes and better understand how the function behaves at extreme values.
• Evaluate how understanding improper rational functions contributes to analyzing limits in calculus.
  • Understanding improper rational functions is essential for evaluating limits in calculus because they often involve approaching infinity or specific discontinuities. By recognizing how these functions behave as they approach their vertical and horizontal asymptotes, students can make informed conclusions about limits and continuity. Additionally, this knowledge aids in determining points of discontinuity and helps clarify how these functions fit into larger analytical frameworks in calculus.

Related terms

rational function: A function that can be expressed as the quotient of two polynomial functions.

asymptote: A line that a graph approaches but never touches, indicating the behavior of the function at extreme values.

horizontal asymptote: A horizontal line that represents the value a rational function approaches as the input approaches infinity or negative infinity.