Rational Functions Examples with Solved Problems

📚 Quick Study Guide

• 🔍 Definition: A rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials. The general form is $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials.
• 📈 Domain: The domain of a rational function is all real numbers except for the values of $x$ where the denominator $Q(x) = 0$. These values are excluded to avoid division by zero.
• ✂️ Simplifying: Rational functions can often be simplified by factoring both the numerator and the denominator and then cancelling out common factors.
• 📍 Vertical Asymptotes: Vertical asymptotes occur at $x$ values where the denominator is zero and the numerator is non-zero after simplification.
• ➖ Horizontal Asymptotes:
  • 🧪 If the degree of $P(x)$ < degree of $Q(x)$, then the horizontal asymptote is $y = 0$.
  • ➗ If the degree of $P(x)$ = degree of $Q(x)$, then the horizontal asymptote is $y = \frac{\text{leading coefficient of } P(x)}{\text{leading coefficient of } Q(x)}$.
  • 🚀 If the degree of $P(x)$ > degree of $Q(x)$, then there is no horizontal asymptote (there may be a slant asymptote).
• ➗ Holes: Holes occur where a factor is cancelled from both the numerator and denominator. The x-coordinate of the hole is the value that makes the cancelled factor zero.

Practice Quiz

1. Which of the following is a rational function?
   1. A. $f(x) = \sqrt{x}$
   2. B. $f(x) = x^2 + 3x - 1$
   3. C. $f(x) = \frac{x+1}{x-2}$
   4. D. $f(x) = |x|$
2. What is the domain of $f(x) = \frac{1}{x+3}$?
   1. A. All real numbers
   2. B. $x \neq 3$
   3. C. $x \neq -3$
   4. D. $x > -3$
3. Simplify the rational function $f(x) = \frac{x^2 - 4}{x - 2}$.
   1. A. $x - 2$
   2. B. $x + 2$
   3. C. $1$
   4. D. Undefined
4. What is the vertical asymptote of $f(x) = \frac{x}{x-5}$?
   1. A. $x = 0$
   2. B. $x = 5$
   3. C. $y = 1$
   4. D. No vertical asymptote
5. What is the horizontal asymptote of $f(x) = \frac{2x^2}{x^2 + 1}$?
   1. A. $y = 0$
   2. B. $y = 1$
   3. C. $y = 2$
   4. D. No horizontal asymptote
6. Where does the hole occur in the function $f(x) = \frac{(x-1)(x+2)}{x-1}$?
   1. A. $x = 1$
   2. B. $x = -2$
   3. C. $x = 0$
   4. D. No hole
7. What is the simplified form of the function $f(x) = \frac{x^2 - 9}{x + 3}$?
   1. A. $x + 3$
   2. B. $x - 3$
   3. C. $x$
   4. D. $\frac{1}{x+3}$