In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. g(x)=(x−3)/(x^2−9)

Key Concepts

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the case of g(x) = (x - 3)/(x^2 - 9), the numerator is a linear polynomial and the denominator is a quadratic polynomial. Understanding the structure of rational functions is essential for analyzing their behavior, including identifying asymptotes and holes.

Vertical Asymptotes

Vertical asymptotes occur in a rational function when the denominator approaches zero while the numerator does not simultaneously approach zero. For g(x), we find vertical asymptotes by setting the denominator, x^2 - 9, equal to zero and solving for x. This helps identify values where the function is undefined and approaches infinity.

Holes in the Graph

Holes in the graph of a rational function occur at values of x where both the numerator and denominator equal zero, indicating a removable discontinuity. In g(x), we need to factor both the numerator and denominator to find common factors. Identifying these holes is crucial for accurately sketching the graph and understanding the function's behavior.