Rational Functions

A rational function is a function that can be expressed as the quotient of two polynomial functions. In other words, it is a function of the form f(x) = p(x) / q(x), where p(x) and q(x) are both polynomial functions and q(x) is not the zero function.

Rational functions can have a variety of properties and behaviors, including asymptotes, intercepts, and holes. Understanding these properties is important for analyzing and graphing rational functions.

Key Concepts

1. Numerator and Denominator: The numerator of a rational function is the polynomial function in the numerator, and the denominator is the polynomial function in the denominator.
2. Vertical Asymptotes: Vertical asymptotes occur at values of x for which the denominator of the rational function is equal to zero. These are vertical lines that the graph approaches but does not cross.
3. Horizontal Asymptotes: Horizontal asymptotes describe the behavior of the function as x approaches positive or negative infinity. The degree of the numerator and denominator polynomials determines the existence and location of horizontal asymptotes.
4. Intercepts: Intercepts of a rational function occur where the graph intersects the x-axis (x-intercepts) or the y-axis (y-intercepts).
5. Holes: Holes in the graph of a rational function occur when factors in the numerator and denominator cancel out, resulting in a gap in the graph at that point.

Study Guide

When studying rational functions, it's important to focus on the following key aspects:

1. Identifying the numerator and denominator of the rational function.
2. Finding vertical asymptotes by setting the denominator equal to zero and solving for x.
3. Determining horizontal asymptotes by comparing the degrees of the numerator and denominator polynomials.
4. Finding x-intercepts by setting the numerator equal to zero and solving for x.
5. Finding y-intercepts by evaluating the function at x=0.
6. Identifying and analyzing any holes in the graph by simplifying the function and determining where it is undefined.
7. Using these properties to sketch the graph of the rational function.

Practice problems involving these aspects of rational functions will help reinforce your understanding and ability to work with these types of functions.

Remember to also pay attention to any restrictions on the domain of the function, as division by zero is not allowed in the real number system.

Good luck with your studies of rational functions!