1. Let's start by stating the problem: What are rational functions? 2. A rational function is a function that can be expressed as the ratio of two polynomials. In other words, it has the form: $$f(x) = \frac{P(x)}{Q(x)}$$ where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$. 3. Important rules: - The denominator polynomial $Q(x)$ cannot be zero because division by zero is undefined. - Rational functions can have vertical asymptotes where $Q(x) = 0$. - They can also have horizontal or oblique asymptotes depending on the degrees of $P(x)$ and $Q(x)$. 4. Example: Consider the rational function $$f(x) = \frac{x^2 - 1}{x - 1}$$ We can factor the numerator: $$x^2 - 1 = (x - 1)(x + 1)$$ So, $$f(x) = \frac{(x - 1)(x + 1)}{x - 1}$$ For all $x \neq 1$, we can simplify: $$f(x) = x + 1$$ However, at $x = 1$, the function is undefined because the denominator is zero. 5. In summary, rational functions are ratios of polynomials with restrictions on the denominator to avoid division by zero. They are important in algebra and calculus for modeling and analyzing behaviors involving ratios of polynomial expressions.