Find the horizontal asymptote, if there is one, of the graph of each rational function. f(x)=(−2x+1)/(3x+5)

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Identify the degrees of the numerator and denominator polynomials in the rational function \(f(x) = \frac{-2x + 1}{3x + 5}\). Here, both the numerator and denominator are first-degree polynomials (degree 1).

Recall 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 \(y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}\). - 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).

Since the degrees of numerator and denominator are equal (both 1), find the leading coefficients: numerator's leading coefficient is \(-2\), denominator's leading coefficient is \(3\).

Write the horizontal asymptote as \(y = \frac{-2}{3}\) by dividing the leading coefficients.

Conclude that the horizontal asymptote of the function \(f(x) = \frac{-2x + 1}{3x + 5}\) is the horizontal line \(y = \frac{-2}{3}\).

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주요 개념

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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.

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.

Degree of Polynomials in Numerator and Denominator

The degrees of the numerator and denominator polynomials determine the horizontal asymptote of a rational function. 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.