Rational Function

Definition

A rational function is a function defined by the ratio of two polynomials, typically expressed as $$f(x) = \frac{P(x)}{Q(x)}$$ where both $$P(x)$$ and $$Q(x)$$ are polynomials and $$Q(x) \neq 0$$. These functions have unique properties such as asymptotes, which influence their graphs, and they can exhibit behaviors like vertical and horizontal asymptotes based on the degrees of the polynomials involved. Understanding rational functions helps in analyzing their behavior, roots, and how they fit within the broader categories of function types.

5 Must Know Facts For Your Next Test

1. Rational functions can have vertical asymptotes where the denominator equals zero, leading to undefined points in the function.
2. The horizontal asymptote is determined by comparing the degrees of the numerator and denominator; if the degree of the numerator is less than that of the denominator, the horizontal asymptote is at y=0.
3. Rational functions can exhibit holes in their graphs, which occur when both the numerator and denominator have common factors that cancel each other out.
4. The domain of a rational function excludes any values that make the denominator zero, so it’s important to identify these values to determine valid inputs.
5. Graphs of rational functions can be used to illustrate limits and continuity, providing insights into their behavior as they approach certain values.

Review Questions

• What role do asymptotes play in understanding the behavior of rational functions?
  • Asymptotes are crucial in illustrating how rational functions behave as they approach certain values or infinity. Vertical asymptotes indicate values where the function is undefined due to division by zero in the denominator. Horizontal asymptotes help determine long-term behavior as x approaches positive or negative infinity, giving insight into whether the function stabilizes at a particular value or diverges.
• How can you identify and analyze holes in a rational function's graph?
  • Holes occur in a rational function's graph when there are common factors in both the numerator and denominator that can be canceled. To identify them, factor both parts of the function and look for shared factors. The x-values at which these factors equal zero represent points on the graph where there is a hole, indicating that while those points are not included in the domain, their limits can still be analyzed to understand nearby behavior.
• Evaluate how changes in polynomial degrees affect the characteristics of rational functions' graphs.
  • Changes in polynomial degrees greatly influence the graph's characteristics. If the degree of the numerator is greater than that of the denominator, the function will typically increase or decrease without bound, impacting horizontal asymptotes. Conversely, if they are equal, the horizontal asymptote corresponds to the ratio of leading coefficients. This analysis reveals critical insights into intersections with axes and overall trends as x varies across its domain.

Related terms

Polynomial Function: 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 are constants and $$n$$ is a non-negative integer.

Asymptote: A line that a graph approaches as it heads towards infinity. For rational functions, vertical asymptotes occur where the denominator equals zero, while horizontal asymptotes relate to the degrees of the numerator and denominator.

Zeros of a Function: The values of $$x$$ for which the function equals zero, found by setting the numerator of a rational function equal to zero and solving for $$x$$.