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2.6 Rational Functions and Asymptotes Rational Function Rational function can be written in the form N ( x) f ( x) D( x) where N(x) and D(x) are polynomials and D(x) is not 0 Domain of a rational function is all real numbers except the x-values that make D(x) = 0 Finding Domain/Range Find the domain and the range of the function 1 f ( x) x x -1 -.5 -.1 -.001 -.0001 f(x) -1 -2 -10 -100 -1000 x .001 .01 .1 .5 1 f(x) 1000 100 10 2 1 Horizontal and Vertical Asymptotes 1) The line x = a is a vertical asymptotes of the graph f if f(x) ∞ or f(x) -∞ as x a, either from the right or from the left 2) The line y = b is a horizontal asymptote of the graph of f if f(x) b as x ∞ or x -∞ Asymptotes of a Rational Function Let f be the rational function where N(x) and D(x) have no common factors. To Find A Vertical Asymptote: The graph has a vertical asymptotes at the zeros of D(x) Asymptotes of a Rational Function Let f be the rational function where N(x) and D(x) have no common factors. To Find A Horizontal Asymptote: The graph of f has at most one horizontal asymptote determined by comparing the degrees of N(x) and D(x) n = the degree of N(x) and d = the degree of D(x) 1. If n < d, then y = 0 2. If n = d, then y = an /bn (leading coefficients) 3. If n > d, then there is no horizontal asymptote Finding the Asymptotes 1) 2x f ( x) 2 3x 1 2) 2x2 f ( x) 2 x 1 Finding Horizontal and Vertical Asymptotes x x2 f ( x) 2 x x6 2 Find the a) domain, b) vertical asymptotes, and c) horizontal asymptotes 3x 3 7 x 2 2 f ( x) 4 x3 5 Two Horizontal Asymptotes A function that is not rational can have two horizontal asymptotes. One to the right and one to the left. x 10 f ( x) x 2 Word Problems A utility company burns coal to generate electricity. The cost C (in $) of removing p% of the smokestack pollutants is given by C=80,000/(100-p) for 0 < p < 100. Use your calculator to graph the function. You are a member of a state legislature that is considering a law that would require utility companies to remove 90% of the pollutants from their smokestack emissions. The current law requires 85% removal. How much additional cost would there be to the utility company because of the new law?