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/(x²-9)
C. ƒ(x)=(x²-9)/(x+3)
D. ƒ(x)=(x+5)/(x+2)(x-3)

Verified step by step guidance

1. Recall that a horizontal asymptote describes the behavior of a function as \(x\) approaches positive or negative infinity. It often depends on the degrees of the polynomials in the numerator and denominator of a rational function.
2. For each function, identify the degree of the numerator and the degree of the denominator:
3. A. \(f(x) = \frac{2x - 7}{x + 3}\): numerator degree is 1, denominator degree is 1.
4. B. \(f(x) = \frac{3x}{x^2 - 9}\): numerator degree is 1, denominator degree is 2.
5. C. \(f(x) = \frac{x^2 - 9}{x + 3}\): numerator degree is 2, denominator degree is 1.
6. D. \(f(x) = \frac{x + 5}{(x + 2)(x - 3)}\): numerator degree is 1, denominator degree is 2 (since denominator expands to a quadratic).
7. Use the rule for horizontal asymptotes of rational functions: If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y=0\). If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (there may be an oblique asymptote instead).
8. Apply this rule to each function to determine which one does not have a horizontal asymptote.

Key Concepts

Horizontal Asymptotes of Rational Functions

A horizontal asymptote describes the behavior of a function as x approaches infinity or negative infinity. For rational functions, it depends on the degrees of the numerator and denominator polynomials. If the degree of the numerator is less than or equal to the degree of the denominator, a horizontal asymptote exists.

Degree of Polynomials in Rational Functions

The degree of a polynomial is the highest power of the variable in the expression. In rational functions, comparing the degrees of numerator and denominator helps determine end behavior and asymptotes. If the numerator's degree is greater than the denominator's, no horizontal asymptote exists.

Simplifying Rational Functions and Identifying Asymptotes

Simplifying rational functions by factoring and canceling common terms can reveal holes or vertical asymptotes. However, horizontal asymptotes depend on degree comparison, not simplification. Understanding this helps in correctly identifying which functions lack horizontal asymptotes.