Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4. ƒ(x)=(4x^2+25)/(x^2+9)

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(x) and Q(x) are polynomials. Understanding the degrees of the polynomials in the numerator and denominator is crucial for analyzing the behavior of the function, particularly in relation to asymptotes.

Asymptotes

Asymptotes are lines that the graph of a function approaches but never touches. There are three types: vertical asymptotes occur where the denominator of a rational function is zero, horizontal asymptotes describe the behavior of the function as x approaches infinity, and oblique (or slant) asymptotes appear 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, the asymptote is the ratio of 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.