1. Introduction to Rational Expressions
Rational expressions are fractions that contain polynomials in the numerator and denominator.
2. Simplifying Rational Expressions
Factorize and cancel out common terms.
Example: \( \frac{x^2 - 9}{x^2 - 6x + 9} = \frac{(x-3)(x+3)}{(x-3)(x-3)} \)
3. Restrictions on the Domain
Find values that make the denominator zero and exclude them.
Example: \( \frac{1}{x+5} \) is undefined at \( x = -5 \).
4. Operations with Rational Expressions
Find the Least Common Denominator (LCD) for addition and subtraction.
Example: Find the LCD of \( \frac{2}{x} + \frac{3}{x+2} \), LCD = \( x(x+2) \)
Exercise: Simplify \( \frac{3}{x+1} + \frac{4}{x-1} \)
5. Solving Equations
Multiply by LCD to eliminate denominators and solve for \( x \).
Example: Solve \( \frac{x+2}{x-3} = \frac{4}{x+3} \)
Exercise: Solve \( \frac{5}{x+2} = \frac{x+3}{x-2} \)
6. Graphing Rational Expressions
Identify vertical and horizontal asymptotes, and plot key points.
Example: Graph \( f(x) = \frac{1}{x-2} \) with vertical asymptote at \( x = 2 \) and horizontal asymptote at \( y = 0 \).
Exercise: Find and graph the asymptotes of \( f(x) = \frac{2x+1}{x-3} \)
📈 Visualization 📈
Graph representation of a rational function: