Rational functions

Definition

Rational functions are functions that can be expressed as the quotient of two polynomial functions. They are important because they help in understanding the behavior of algebraic curves and play a crucial role in studying morphisms between algebraic varieties, showcasing how one can relate different geometric objects through algebraic expressions.

5 Must Know Facts For Your Next Test

1. A rational function can be written in the form $$f(x) = \frac{P(x)}{Q(x)}$$ where both P(x) and Q(x) are polynomials.
2. The domain of a rational function excludes points where the denominator Q(x) equals zero, as these points lead to undefined values.
3. Rational functions can have vertical asymptotes at the roots of the denominator and horizontal asymptotes that describe end behavior based on the degrees of P(x) and Q(x).
4. They can be manipulated through algebraic operations such as addition, subtraction, multiplication, and division, following specific rules to maintain their rational form.
5. In the context of morphisms, rational functions can help establish connections between different varieties by defining maps that reflect their geometric properties.

Review Questions

• How do rational functions provide insights into the behavior of algebraic curves?
  • Rational functions allow us to understand the behavior of algebraic curves by providing a way to represent curves as quotients of polynomials. This representation reveals critical information about intersections, asymptotic behavior, and singularities. By analyzing these functions, we can glean information about how curves behave at various points, especially near poles and zeros.
• Discuss how the concept of rational functions relates to morphisms in algebraic geometry.
  • Rational functions are instrumental in defining morphisms between algebraic varieties. A morphism can often be represented using rational functions that map points from one variety to another while preserving structure. This relationship helps in studying how different varieties relate to each other through algebraic expressions and provides a framework for understanding transformations in a geometric context.
• Evaluate the implications of vertical and horizontal asymptotes in the context of rational functions and their morphisms.
  • Vertical and horizontal asymptotes in rational functions have significant implications for both their graphical representation and their role in morphisms. Vertical asymptotes indicate where the function is undefined, impacting the mapping between varieties and showing where certain properties may change abruptly. Horizontal asymptotes reflect long-term behavior as inputs approach infinity, which can influence how a morphism behaves over large scales. Analyzing these aspects helps understand continuity, limits, and overall structure in the relationships defined by these functions.

Related terms

Polynomial function: A function that is represented by a polynomial, which is an expression made up of variables raised to non-negative integer powers and combined using addition, subtraction, and multiplication.

Algebraic varieties: Geometric objects that are defined as the solution sets of systems of polynomial equations, serving as fundamental objects of study in algebraic geometry.

Morphisms:

Functions between algebraic varieties that preserve their structure, allowing for the mapping of points from one variety to another while maintaining their algebraic relationships.