Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4. ƒ(x)=(4-3x)/(2x+1)

Key Concepts

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials, typically in the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials. Understanding the structure of rational functions is essential for analyzing their behavior, particularly in relation to asymptotes.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. There are three types: vertical asymptotes occur where the function is undefined (typically where the denominator is zero), horizontal asymptotes describe the behavior of the function as x approaches infinity, and oblique (or slant) asymptotes occur when the degree of the numerator is one greater than that of the denominator.

Finding Asymptotes

To find vertical asymptotes, set the denominator equal to zero and solve for x. For horizontal asymptotes, compare the degrees of the numerator and denominator: if they are equal, divide the leading coefficients; if the numerator's degree is less, the asymptote is y=0; if greater, there is no horizontal asymptote. Oblique asymptotes can be found using polynomial long division when the numerator's degree exceeds that of the denominator by one.