Rational Functions

Definition

Rational functions are mathematical expressions that can be represented as the ratio of two polynomial functions. These functions are significant because they can model various real-world situations and are often analyzed in relation to their limits, especially when approaching points where the function is undefined or exhibits vertical asymptotes.

5 Must Know Facts For Your Next Test

1. Rational functions are defined as the quotient of two polynomials, expressed as $$f(x) = \frac{P(x)}{Q(x)}$$, where P and Q are polynomial functions.
2. The domain of a rational function is restricted by the values that make the denominator zero, as these points lead to undefined behavior.
3. Vertical asymptotes occur at the values of x that make the denominator zero while the numerator is non-zero at those points.
4. Horizontal asymptotes can exist in rational functions, which describe the behavior of the function as x approaches infinity; these depend on the degrees of the numerator and denominator.
5. Analyzing limits of rational functions is crucial for understanding their behavior near points of discontinuity and helps determine the existence of asymptotes.

Review Questions

• How do you determine the vertical asymptotes of a rational function?
  • To find vertical asymptotes of a rational function, you need to identify values of x that make the denominator equal to zero while ensuring that the numerator is not also zero at those points. Essentially, set the denominator Q(x) to zero and solve for x. The resulting x-values indicate where the function approaches infinity, leading to vertical asymptotes on the graph.
• Discuss how horizontal asymptotes are determined in rational functions based on the degrees of their polynomials.
  • Horizontal asymptotes in rational functions are determined by comparing the degrees of the numerator and denominator polynomials. If the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote at y = 0. If the degrees are equal, the horizontal asymptote is at y = \frac{a}{b}, where a and b are the leading coefficients of the numerator and denominator respectively. If the numerator's degree is greater than that of the denominator, there is no horizontal asymptote.
• Evaluate how understanding limits involving rational functions can affect predictions about their behavior near discontinuities.
  • Understanding limits with rational functions allows us to predict how they behave near points where they may be discontinuous, such as vertical asymptotes. By calculating limits from both sides of a point where Q(x) equals zero, we can ascertain whether the function tends towards positive or negative infinity, which informs us about its overall behavior in that vicinity. This analysis helps in sketching accurate graphs and understanding real-world scenarios modeled by these functions.

Related terms

Polynomial Function: A function that can be expressed as a sum of terms, each consisting of a variable raised to a non-negative integer power multiplied by a coefficient.

Asymptote: A line that a graph approaches as the values of the variable increase or decrease without bound; rational functions may have vertical and horizontal asymptotes.

Limit: A fundamental concept in calculus that describes the behavior of a function as its argument approaches a particular value.