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)

Key Concepts

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. The general form is f(x) = P(x)/Q(x), where P and Q are polynomials. Understanding the degrees of the polynomials in the numerator and denominator is crucial for analyzing the behavior of the function, especially as x approaches infinity.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches positive or negative infinity. For rational functions, the horizontal asymptote can be determined by comparing the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0; if they are equal, the asymptote is y = leading coefficient of P / leading coefficient of Q.

Leading Coefficients

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. In the context of rational functions, the leading coefficients of the numerator and denominator are essential for determining the horizontal asymptote when the degrees of the polynomials are equal. This concept helps simplify the analysis of the function's end behavior.