Graph each rational function. ƒ(x)=(3x²+3x-6)/(x²-x-12)

Verified step by step guidance

1. Identify the rational function given: \(f(x) = \frac{3x^2 + 3x - 6}{x^2 - x - 12}\).
2. Factor both the numerator and the denominator to simplify the function and find key features. For the numerator, factor \$3x^2 + 3x - 6\(, and for the denominator, factor \)x^2 - x - 12$.
3. Determine the domain by finding the values of \(x\) that make the denominator zero, since these values are excluded from the domain and correspond to vertical asymptotes or holes.
4. Find the vertical asymptotes by setting the denominator equal to zero and solving for \(x\). Also, check if any factors cancel with the numerator to identify holes instead of vertical asymptotes.
5. Find the horizontal or oblique asymptote by comparing the degrees of the numerator and denominator polynomials. Since both are degree 2, divide the leading coefficients to find the horizontal asymptote.

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x), where Q(x) ≠ 0. Understanding the behavior of rational functions involves analyzing their numerator and denominator to identify domain restrictions, zeros, and asymptotes.

Domain and Vertical Asymptotes

The domain of a rational function excludes values that make the denominator zero. These values often correspond to vertical asymptotes, where the function approaches infinity or negative infinity, indicating points of discontinuity on the graph.

Horizontal and Oblique Asymptotes

Horizontal asymptotes describe the end behavior of a rational function as x approaches infinity or negative infinity. They depend on the degrees of the numerator and denominator polynomials. If the numerator's degree is less than or equal to the denominator's, a horizontal asymptote exists; otherwise, an oblique asymptote may occur.