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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 39
Chapter 4, Problem 39

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

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Identify the degrees of the numerator and denominator polynomials in the rational function \(g(x) = \frac{12x^2}{3x^2 + 1}\). The degree of the numerator is 2, and the degree of the denominator is also 2.
Recall the rule for horizontal asymptotes of rational functions: If the degrees of numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients.
Determine the leading coefficient of the numerator, which is 12 (from \$12x^2\(), and the leading coefficient of the denominator, which is 3 (from \)3x^2$).
Write the horizontal asymptote as \(y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}} = \frac{12}{3}\).
Simplify the fraction to express the horizontal asymptote in simplest form.

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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, especially their graphs, is essential for analyzing asymptotes and limits.
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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 of Polynomials in Rational Functions

The degrees of the numerator and denominator polynomials determine the horizontal asymptote. If degrees are equal, the asymptote is the ratio of leading coefficients; if numerator degree is less, asymptote is y=0; if greater, no horizontal asymptote exists.
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Intro to Rational Functions
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