In Exercises 37–44, 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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1. Identify the degrees of the numerator and denominator polynomials in the rational function $f\left(x\right)=\frac{-2x+1}{3x+5}$. Both numerator and denominator are first-degree polynomials (degree 1).
2. Recall that the horizontal asymptote depends on the degrees of numerator and denominator: if degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.
3. Identify the leading coefficient of the numerator, which is $-2$, and the leading coefficient of the denominator, which is $3$.
4. Write the horizontal asymptote as the ratio of these leading coefficients: $y\; =\; \backslash frac\{-2\}\{3\}$.
5. Conclude that the horizontal asymptote of the function $f(x)$ is the horizontal line $y\; =\; \backslash frac\{-2\}\{3\}$.

Key Concepts

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

Horizontal asymptotes describe the behavior of a function as x approaches infinity or negative infinity. For rational functions, they indicate the value that f(x) approaches, determined by comparing the degrees of the numerator and denominator polynomials.

Degree Comparison of Polynomials

The degrees of the numerator and denominator polynomials dictate the horizontal asymptote of a rational function. If degrees are equal, the asymptote is the ratio of leading coefficients; if numerator degree is less, the asymptote is y=0; if greater, no horizontal asymptote exists.