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Rational Functions Overview

Sep 16, 2025,

Overview

This lecture covers the definition, properties, and key features of rational functions, focusing on finding domains, ranges, asymptotes, and real-world applications.

Defining Rational Functions

  • A rational function is a function that is the ratio of two polynomials.
  • Rational functions are typically written as f(x) = p(x)/q(x), where q(x) ≠ 0.
  • The denominator cannot be zero because division by zero is undefined, creating restrictions on the domain.

Examples and Notation

  • Example: f(x) = (numerator polynomial)/(denominator polynomial), e.g., f(x) = 3x/(x³ - 2x² - 3x).
  • Always include function notation (e.g., f(x)=...) and specify restrictions on x.

Domain and Vertical Asymptotes

  • To find the domain, set the denominator equal to zero and solve for x; these values are excluded from the domain.
  • Use set notation for the domain: all real numbers except where the denominator is zero.
  • Vertical asymptotes (VA) are at x-values that make the denominator zero, after cancelling any common factors.
  • Cancelled factors create holes (discontinuities), not vertical asymptotes.

Factoring and Finding Asymptotes (Example)

  • Factor the denominator to find zeros (e.g., x³ - 2x² - 3x factors to x(x+1)(x-3)).
  • Solve x = 0, x = -1, and x = 3; these restrict the domain.
  • If a factor cancels with the numerator, it’s a hole (not a VA); remaining factors give VAs.

Horizontal Asymptotes and Range

  • Horizontal asymptote (HA) depends on the degrees (highest exponents) of numerator and denominator polynomials:
    • If degree numerator < degree denominator, HA at y = 0.
    • If degrees are equal, HA at y = (leading coefficient numerator)/(leading coefficient denominator).
    • If degree numerator > degree denominator, no HA.
  • If degree of numerator is exactly one more than denominator, there’s a slant asymptote.
  • The range of a rational function is all real y-values except the y-value of the HA.

Applying Asymptote Rules (Exam Practice)

  • For a specified HA and VA, use their rules to determine correct rational functions.
  • To create factors for VAs, set VA values as solutions: x = a → factor is (x - a).

Key Terms & Definitions

  • Rational Function — a function that is a ratio of two polynomials.
  • Domain — all real x-values except those that make the denominator zero.
  • Vertical Asymptote (VA) — a vertical line where the function approaches infinity, based on denominator zeros not cancelled by the numerator.
  • Horizontal Asymptote (HA) — a horizontal line y = c that the function approaches; determined by polynomial degrees and coefficients.
  • Discontinuity — a hole in the graph caused by cancelled factors.
  • Slant Asymptote — occurs when numerator degree is one more than denominator degree.

Action Items / Next Steps

  • Practice factoring denominators and finding domain, VA, HA for given rational functions.
  • Review set notation for expressing domains.
  • Complete assigned homework problems similar to class examples.
  • Prepare for mid-module and module exams on rational functions.

Full transcript