Rational Functions

Rational Functions

Definition

• Rational functions are functions that can be expressed as the quotient of two polynomial functions.

Structure

• A rational function is of the form: $R(x) = \frac{P(x)}{Q(x)}$ where:
  • $P(x)$ is a polynomial in the numerator.
  • $Q(x)$ is a polynomial in the denominator.

Properties

• Domain: The domain of a rational function excludes values of $x$ that make $Q(x) = 0$ (denominator cannot be zero).
• Asymptotes:
  • Vertical asymptotes occur at values where $Q(x) = 0$.
  • Horizontal asymptotes are determined by the degrees of the numerator and denominator polynomials:
  • If the degree of $P$ is less than that of $Q$, the horizontal asymptote is at $y = 0$.
  • If the degree of $P$ is equal to that of $Q$, the horizontal asymptote is at $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of $P$ and $Q$, respectively.
  • If the degree of $P$ exceeds that of $Q$ by one, the function will have an oblique asymptote.

Holes

• Holes in rational functions are points at which the function is undefined but does not have vertical asymptotes.
  • Occurrence: Holes occur where both the numerator and denominator have a common factor that can be canceled out.
  • Example: If $R(x) = \frac{(x-3)(x+1)}{(x-3)(x-2)}$, there is a hole at $x = 3$ because this factor cancels out.

Example

• Consider the rational function: $R(x) = \frac{x^2 - 4}{x^2 - x - 6}$
  • Factor the numerator and denominator:
  • $x^2 - 4 = (x-2)(x+2)$
  • $x^2 - x - 6 = (x-3)(x+2)$
  • Simplified Form:
  • $R(x) = \frac{(x-2)(x+2)}{(x-3)(x+2)}$
  • Identify holes at $x = -2$ (hole occurs at the cancellation) and vertical asymptote at $x = 3$.

Conclusion

• Understanding the characteristics of rational functions, including their domain, asymptotes, and holes, is crucial in analyzing their behavior.