Rational functions test pdf

📚 Quick Study Guide

• 🔍 A rational function is a function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials.
• ➗ The general form of a rational function is $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.
• 📈 Vertical asymptotes occur at $x$ values where the denominator $Q(x) = 0$, provided the numerator $P(x)$ is not also zero at that point.
• ➖ Holes occur when both the numerator and denominator are zero at the same $x$ value. Simplify the rational function to remove the common factor causing the hole.
• ↔️ Horizontal asymptotes are 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 $Q(x)$, then there is no horizontal asymptote (but there may be a slant asymptote).
• ✏️ To find the $x$-intercepts, set $P(x) = 0$ and solve for $x$.
• 📍 To find the $y$-intercept, evaluate $f(0)$, i.e., $f(0) = \frac{P(0)}{Q(0)}$.

🧪 Practice Quiz

1. What is the vertical asymptote of the rational function $f(x) = \frac{x+2}{x-3}$?
   1. x = -2
   2. x = 2
   3. x = -3
   4. x = 3
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 rational functions has a hole at $x = 1$?
   1. $f(x) = \frac{x+1}{x-1}$
   2. $f(x) = \frac{x-1}{x+1}$
   3. $f(x) = \frac{(x-1)(x+2)}{x-1}$
   4. $f(x) = \frac{x+2}{x-1}$
4. What is the $x$-intercept of the rational function $f(x) = \frac{x-5}{x+2}$?
   1. x = -2
   2. x = 2
   3. x = -5
   4. x = 5
5. What is the $y$-intercept of the rational function $f(x) = \frac{x+3}{x-1}$?
   1. y = -3
   2. y = 3
   3. y = -1
   4. y = -3
6. What is the domain of the rational function $f(x) = \frac{1}{x^2-9}$?
   1. All real numbers
   2. $x \neq 3$
   3. $x \neq -3$
   4. $x \neq 3$ and $x \neq -3$
7. What is the oblique (slant) asymptote of the rational function $f(x) = \frac{x^2+2x+1}{x-1}$?
   1. y = x + 1
   2. y = x + 3
   3. y = x - 1
   4. There is no oblique asymptote