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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 37
Chapter 4, Problem 37

Find the horizontal asymptote, if there is one, of the graph of each rational function. f(x)=12x/(3x2+1)

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Identify the degrees of the numerator and denominator polynomials in the rational function \(f(x) = \frac{12x}{3x^{2} + 1}\).
Recall that the degree of the numerator is 1 (since \$12x\( is a first-degree polynomial) and the degree of the denominator is 2 (since \)3x^{2}$ is a second-degree polynomial).
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 \(y = 0\).
Since the degree of the numerator (1) is less than the degree of the denominator (2), conclude that the horizontal asymptote is \(y = 0\).
Write the final conclusion that the graph of \(f(x)\) has a horizontal asymptote at \(y = 0\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding the behavior of rational functions involves analyzing the degrees and coefficients of the numerator and denominator polynomials.
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Horizontal Asymptotes

A horizontal asymptote describes the behavior of a function as x approaches infinity or negative infinity. It is a horizontal line y = L that the graph approaches but does not necessarily touch, indicating the end behavior of the function.
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Degree Comparison of Polynomials

To find horizontal asymptotes of rational functions, compare the degrees of the numerator and denominator polynomials. If the numerator's degree is less, the asymptote is y=0; if equal, it is the ratio of leading coefficients; if greater, there is no horizontal asymptote.
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