Rational Function

Definition

A rational function is a function that can be expressed as the ratio of two polynomial functions. This means that it takes the form $$f(x) = \frac{P(x)}{Q(x)}$$, where both P(x) and Q(x) are polynomials and Q(x) is not equal to zero. Understanding rational functions is important because they can exhibit unique behaviors such as asymptotes and discontinuities, and they also play a significant role in various mathematical applications.

5 Must Know Facts For Your Next Test

1. Rational functions can have vertical asymptotes at values of x where the denominator equals zero, provided that the numerator does not also equal zero at those points.
2. Horizontal asymptotes in rational functions are determined by comparing the degrees of the numerator and denominator polynomials: if the degree of P is less than Q, the horizontal asymptote is at y=0; if they are equal, it is at $$y = \frac{a}{b}$$, where a and b are the leading coefficients.
3. The domain of a rational function consists of all real numbers except for those values that make the denominator equal to zero.
4. Rational functions can have holes in their graphs, which occur when both the numerator and denominator share a common factor that cancels out.
5. In real-world applications, rational functions can model scenarios such as rates, proportions, and certain types of growth or decay processes.

Review Questions

• How do vertical asymptotes affect the behavior of a rational function's graph?
  • Vertical asymptotes indicate where the function's value approaches infinity or negative infinity. They occur at x-values that make the denominator zero while keeping the numerator non-zero. The presence of these asymptotes can lead to discontinuities in the graph, causing it to break or jump as it gets closer to those x-values.
• What is the relationship between the degrees of the polynomials in the numerator and denominator regarding horizontal asymptotes?
  • The degrees of the polynomials in rational functions play a crucial role in determining horizontal asymptotes. If the degree of the numerator polynomial is less than that of the denominator, the horizontal asymptote is at y=0. If they are equal, the horizontal asymptote will be at y equal to the ratio of their leading coefficients. If the degree of the numerator exceeds that of the denominator, there will be no horizontal asymptote.
• Evaluate how discontinuities in rational functions can impact their overall characteristics and interpretations in applied mathematics.
  • Discontinuities in rational functions indicate points where a function cannot be defined or where its value drastically changes. This has significant implications in applied mathematics since these points can represent limits or thresholds in real-world scenarios, such as maximum capacities or breakpoints in relationships. Understanding these discontinuities allows for better modeling and interpretation of data trends and behaviors, ensuring accurate predictions and analyses.

Related terms

Polynomial Function: A polynomial function is a function that can be expressed in the form $$P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$, where the coefficients $$a_i$$ are constants and n is a non-negative integer.

Asymptote: An asymptote is a line that a graph approaches but never touches. In the context of rational functions, there can be vertical, horizontal, or oblique asymptotes.

Discontinuity: A discontinuity occurs at a point in a function where it is not continuous. For rational functions, this often happens at points where the denominator is zero.