Rational Functions

Definition

Rational functions are mathematical expressions formed by the ratio of two polynomials. They can be expressed in the form $$R(x) = \frac{P(x)}{Q(x)}$$ where $$P(x)$$ and $$Q(x)$$ are polynomials, and $$Q(x) \neq 0$$. Rational functions are important in understanding field extensions, as they illustrate how elements can be combined and manipulated within various fields.

5 Must Know Facts For Your Next Test

1. Rational functions can have vertical asymptotes where the denominator equals zero, indicating points where the function is undefined.
2. The domain of a rational function excludes values that make the denominator zero, which is critical in determining its behavior.
3. Rational functions can be simplified by factoring out common terms in the numerator and denominator, revealing potential holes or removable discontinuities.
4. The behavior of rational functions at infinity is determined by the degrees of the numerator and denominator polynomials, leading to horizontal or oblique asymptotes.
5. Rational functions are used in various mathematical contexts, including calculus and algebra, often as examples of how to find limits or analyze behaviors near critical points.

Review Questions

• How do rational functions demonstrate properties of field extensions, particularly in terms of element manipulation?
  • Rational functions illustrate properties of field extensions by showing how elements from one field can be combined to form new elements in a larger field. Specifically, when dealing with rational functions, we can view the set of rational functions as a field itself, where addition and multiplication are defined. This allows for the exploration of how roots of polynomials relate to field extensions and how we can construct new fields by adding elements that satisfy certain polynomial equations.
• Discuss the significance of vertical asymptotes and holes in rational functions regarding their domain and continuity.
  • Vertical asymptotes occur at values where the denominator of a rational function is zero, indicating that the function approaches infinity at these points. In contrast, holes represent values where both the numerator and denominator are zero after simplification, leading to removable discontinuities. Understanding these features is crucial for analyzing the function's domain, as they define where the function is undefined and impact its continuity across its entire range.
• Evaluate how the concepts of horizontal asymptotes and polynomial degree influence the behavior of rational functions at infinity.
  • The concepts of horizontal asymptotes arise from examining the degrees of the numerator and denominator polynomials in rational functions. If the degree of the numerator is less than that of the denominator, the horizontal asymptote is at $$y=0$$. If they are equal, the asymptote is given by the ratio of their leading coefficients. If the numerator's degree exceeds that of the denominator, there is no horizontal asymptote; instead, there may be an oblique asymptote. Analyzing these behaviors provides insights into how rational functions behave as their inputs grow very large or very small, which is essential in calculus for understanding limits.

Related terms

Polynomial: An expression that consists of variables raised to non-negative integer powers and coefficients, combined using addition, subtraction, and multiplication.

Field: A set equipped with two operations (addition and multiplication) that satisfy certain properties, allowing for the manipulation of elements within that set.

Algebraic Closure: A field extension in which every non-constant polynomial has a root, meaning that any polynomial equation can be solved within this larger field.