2.6

A rational function is of the form f(x) = where N(x) and D(x) are polynomials and D(x) is NOT the zero polynomial. The domain of the rational function includes all real numbers except x-values that make the denominator zero. Rational Function

Find the domain of f(x) = and discuss the behavior of f near any excluded x-values. Example 1: Finding the Domain of a Rational Function

Domain (-∞, 0) U (0, ∞) Range (-∞, 0) U (0, ∞) No Intercepts Decreasing on (-∞, 0) and (0, ∞) Odd Function Vertical Asymptote at y-axis Horizontal Asymptote at x-axis Origin Symmetry Basic Characteristics of f(x) =

1.The line x = a is a vertical asymptote of the graph of f if f(x) ∞ or f(x) -∞ as x a, either from the left or from the right. 2.The line y = b is a horizontal asymptote of the graph of f if f(x) b as x ∞ or x -∞. Definition of Vertical & Horizontal Asymptotes

Let f be the rational function where N(x) and D(x) have no common factors. 1.The graph of f has vertical asymptotes at the zeros of D(x). 2.The graph of f has at most one horizontal asymptote determined by comparing the degrees of N(x) and D(x). Asymptotes of a Rational Function

A.If n < m, the graph of f has the line y = o (the x-axis) as a horizontal asymptote. B.If n = m the graph of f has the line y = as a horizontal asymptote, where a n is the leading coefficient of the numerator and b m is the leading coefficient of the denominator. C.If n > m, the graph of f has no horizontal asymptote. Horizontal Asymptotes Continued

Find all horizontal and vertical asymptotes of the graph of the rational function f(x) = Example 2: Finding Horizontal & Vertical Asymptotes

For the function f, find A) the domain and B) the vertical asymptotes of f and C) the horizontal asymptotes of f. f(x) = Example 3: Finding a Function’s Domain & Asymptotes

A function that is not rational can have two horizontal asymptotes. Example 4: A Graph with Two Horizontal Asymptotes