Rational functions are an essential topic in the study of algebra. They are expressed as a ratio of two polynomial functions, with the denominator not equal to zero. Let's take a look at an example:

Example 1: Consider the rational function f(x) = (2x^2 + 3x - 1) / (x - 2). In this function, the numerator is a quadratic polynomial, and the denominator is a linear polynomial.

To determine the domain of a rational function, we need to exclude any values of x that would make the denominator equal to zero. In this example, x cannot be equal to 2, as it would result in a division by zero. Thus, the domain of f(x) is all real numbers except x = 2.

Vertical asymptotes are vertical lines that the graph of a rational function approaches as x tends to positive or negative infinity. For this example, the vertical asymptote occurs at x = 2, which aligns with the excluded value from the domain.

By understanding rational functions and their properties, we can solve various real-world problems and gain deeper insights into mathematical concepts.