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Rational Functions - College Algebra

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이 집합의 용어 (19)

  • What is a rational function?
    A function f defined by \(f(x)=\frac{p(x)}{q(x)}\) where p(x) and q(x) are polynomials and q(x) ≠ 0.
  • What does the notation \(x \to c^+\) mean?
    x approaches c from the right (values greater than c but not equal to c).
  • What does the notation \(x \to c^-\) mean?
    x approaches c from the left (values less than c but not equal to c).
  • Define a vertical asymptote of a function f.
    The line x = c is a vertical asymptote if f(x) approaches infinity or negative infinity as x approaches c from either side.
  • How to find vertical asymptotes of a rational function \(f(x)=\frac{p(x)}{q(x)}\)?
    Find values of x where q(x) = 0 but p(x) ≠ 0, assuming p and q have no common factors.
  • What is a horizontal asymptote of a function f?
    The line y = d is a horizontal asymptote if f(x) approaches d as x approaches infinity or negative infinity.
  • Horizontal asymptote rule when degree of numerator n < degree of denominator m?
    The horizontal asymptote is y = 0 (the x-axis).
  • Horizontal asymptote rule when degree of numerator n = degree of denominator m?
    The horizontal asymptote is y = \(\frac{a_n}{b_m}\), the ratio of leading coefficients.
  • Horizontal asymptote rule when degree of numerator n > degree of denominator m?
    There is no horizontal asymptote.
  • Can the graph of a rational function cross its vertical asymptote?
    No, the graph cannot cross a vertical asymptote.
  • Can the graph of a rational function cross its horizontal asymptote?
    Yes, the graph may cross a horizontal asymptote.
  • What is a slant (oblique) asymptote?
    A slant asymptote occurs if the degree of the numerator is exactly one greater than the degree of the denominator.
  • How to find the equation of a slant asymptote?
    Divide the numerator by the denominator; the quotient (linear) is the slant asymptote y = quotient.
  • List the steps to graph a rational function.
    1. Check symmetry. 2. Find y-intercept. 3. Find x-intercepts. 4. Find vertical asymptotes. 5. Find horizontal asymptotes. 6. Plot points between and beyond intercepts and asymptotes. 7. Use info to sketch graph.
  • What is the horizontal asymptote of \(f(x)=\frac{2x^3-3}{9x^2}\)?
    No horizontal asymptote because degree of numerator (3) > degree of denominator (2).
  • What is the horizontal asymptote of \(f(x)=\frac{2x^2-3}{9x^2}\)?
    y = \(\frac{2}{9}\) because degrees are equal and leading coefficients are 2 and 9.
  • What is the horizontal asymptote of \(f(x)=\frac{3x^2-2}{9x^3}\)?
    y = 0 because degree of numerator (2) < degree of denominator (3).
  • In Young's rule dosage formula \(C(x)=\frac{12d x}{x+13}\), what is the horizontal asymptote and its meaning?
    Horizontal asymptote is y = 12d, meaning the dosage approaches 12 times the adult dosage as age increases.
  • In Young's rule dosage formula, what is the vertical asymptote?
    Vertical asymptote at x = -13, which is outside the domain since age x ≥ 2.