Work each problem. Which function has a graph that does not have a horizontal asymptote? A. ƒ(x)=(2x-7)/(x+3) B. ƒ(x)=3x/(x^2-9) C. ƒ(x)=(x^2-9)/(x+3) D. ƒ(x)=(x+5)/(x+2)(x-3)

Verified step by step guidance

1. Step 1: Understand the concept of horizontal asymptotes. Horizontal asymptotes occur when the degree of the polynomial in the numerator is less than or equal to the degree of the polynomial in the denominator.
2. Step 2: Analyze function A, \( f(x) = \frac{2x-7}{x+3} \). The degree of the numerator is 1 and the degree of the denominator is 1. Since the degrees are equal, there is a horizontal asymptote.
3. Step 3: Analyze function B, \( f(x) = \frac{3x}{x^2-9} \). The degree of the numerator is 1 and the degree of the denominator is 2. Since the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote.
4. Step 4: Analyze function C, \( f(x) = \frac{x^2-9}{x+3} \). The degree of the numerator is 2 and the degree of the denominator is 1. Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
5. Step 5: Analyze function D, \( f(x) = \frac{x+5}{(x+2)(x-3)} \). The degree of the numerator is 1 and the degree of the denominator is 2. Since the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote.

Key Concepts

Horizontal Asymptotes

A horizontal asymptote is a horizontal line that a graph approaches as the input values (x) approach positive or negative infinity. It indicates the behavior of a function at extreme values. To determine the presence of a horizontal asymptote, one typically compares the degrees of the polynomial in the numerator and denominator of a rational function.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. They can exhibit various behaviors, including vertical and horizontal asymptotes, depending on the degrees of the polynomials involved. Understanding the structure of rational functions is crucial for analyzing their graphs and identifying asymptotic behavior.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. In the context of rational functions, the degrees of the numerator and denominator determine the existence and type of asymptotes. For example, if the degree of the numerator is greater than that of the denominator, the function does not have a horizontal asymptote.