Rational Functions and Equations quiz

📚 Quick Study Guide

• 🔍 Rational Function Definition: A rational function is a function that can be written as the ratio of two polynomials, $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$.
• 📈 Vertical Asymptotes: Vertical asymptotes occur at $x$-values where the denominator $Q(x) = 0$, but the numerator $P(x) \neq 0$.
• 📉 Horizontal Asymptotes: Determined by comparing the degrees of $P(x)$ and $Q(x)$.
  • 🧪 If degree of $P(x)$ < degree of $Q(x)$, then $y = 0$ is the horizontal asymptote.
  • ➗ If degree of $P(x)$ = degree of $Q(x)$, then $y = \frac{\text{leading coefficient of } P(x)}{\text{leading coefficient of } Q(x)}$ is the horizontal asymptote.
  • 🚀 If degree of $P(x)$ > degree of $Q(x)$, there is no horizontal asymptote (but there may be an oblique asymptote).
• 🧩 Holes: Holes occur at $x$-values where both $P(x) = 0$ and $Q(x) = 0$ after simplification.
• 💡 Solving Rational Equations: To solve rational equations, multiply both sides by the least common denominator (LCD) to eliminate fractions, then solve the resulting equation. Be sure to check for extraneous solutions!
• 📝 Extraneous Solutions: These are solutions obtained during the solving process that do not satisfy the original equation. Always check your answers!
• ➗ Inverse Variation: A relationship where one variable increases as the other decreases, described by the equation $y = \frac{k}{x}$, where $k$ is a constant.

Practice Quiz

1. What is the vertical asymptote of the rational function $f(x) = \frac{x+2}{x-3}$?
   1. $x = -2$
   2. $x = 3$
   3. $y = 1$
   4. $y = 0$
2. What is the horizontal asymptote of the rational function $f(x) = \frac{2x^2 + 1}{x^2 - 4}$?
   1. $y = 0$
   2. $y = 1$
   3. $y = 2$
   4. No horizontal asymptote
3. Which of the following describes the solution to the rational equation $\frac{1}{x} + \frac{1}{2x} = 3$?
   1. $x = \frac{1}{2}$
   2. $x = \frac{1}{3}$
   3. $x = \frac{1}{4}$
   4. No solution
4. What is the domain of the function $f(x) = \frac{x}{x^2 - 9}$?
   1. All real numbers
   2. $x \neq 0$
   3. $x \neq 3, x \neq -3$
   4. $x > 3$
5. Where does the function $f(x) = \frac{(x-2)(x+1)}{(x-2)}$ have a hole?
   1. $x = -1$
   2. $x = 2$
   3. $x = 0$
   4. No hole
6. Solve for $x$: $\frac{x}{x+1} = \frac{2}{3}$
   1. $x = 1$
   2. $x = 2$
   3. $x = 3$
   4. $x = 4$
7. Which equation represents inverse variation?
   1. $y = 3x$
   2. $y = x + 5$
   3. $y = \frac{5}{x}$
   4. $y = x^2$