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AP Precalculus: Rational Functions Formulas

1. Rational Function Form

General form:   \( f(x) = \frac{P(x)}{Q(x)} \)
where \(P(x)\), \(Q(x)\) are polynomials, \(Q(x) \neq 0\)

  • Excluded values: All \(x\) for which \(Q(x)=0\)
  • Domain: \( x \neq \) values that make denominator zero

2. Asymptotes

  • Vertical Asymptotes: Set \(Q(x)=0\) (after canceling common factors)
  • Horizontal Asymptote:
    • - If degree \( P < Q \), y = 0
    • - If degree \( P = Q \), \(y = \frac{\text{leading coefficient of }P}{\text{leading coefficient of }Q}\)
    • - If degree \( P > Q \), no horizontal (see Oblique)
  • Oblique (Slant) Asymptote: If degree \( P = Q + 1 \), use polynomial division: \( f(x) = mx + b + \frac{R(x)}{Q(x)} \); asymptote is \( y = mx + b \)

3. Solving Rational Equations

  1. Find LCD (Least Common Denominator)
  2. Multiply both sides by LCD to clear denominators
  3. Solve resulting equation
  4. Check for extraneous solutions (must not make denominator zero)
Example: \(\frac{1}{x} + 2 = \frac{3}{x}\)
LCD is \(x\), so \( 1 + 2x = 3 \implies x = 1 \)

4. Check Whether Two Rational Functions Are Inverses

Functions \(f(x)\) and \(g(x)\) are inverses if:
\( f(g(x)) = x \) and \( g(f(x)) = x \)
(for all \(x\) in the domains of \(g\) and \(f\), respectively)