Pre Calculus Rational Functions & Asymptotes Rational Function Rational functions are quotients of polynomial functions. This means rational ( ) functions can be expressed as ( ) where p and q are polynomial ( ) functions and q(x) is not zero. The domain of a rational function is the set of all real numbers except the x-values that make the denominator zero. Example: x 5 (a) Find the domain of fx . x2 25 ( ) x2 25 (b) Find the domain of fx . x 5 ( ) ( ) x (c) Find the domain of gx . x2 25 ( ) ( ) ( ) over Pre Calculus (2.6) Rational Functions & Asymptotes Notes on Notation Symbol Meaning Symbol Meaning xa x approaches a from the right side xa x approaches a from the left side x x approaches infinity x x approaches negative infinity Definition of a Vertical Asymptote The line xa is a vertical asymptote of the graph of a function f if fx increases or decreases without bound as x approaches a. Symbolically this means that, fx() as xa or as xa . Locating Vertical Asymptotes px If fx is a rational function in which px and qx have no common qx factors and a is a zero of qx , the denominator, then xa is a vertical asymptote of the graph of f. CHECK THE RATIONAL FUNCTION FOR HOLES (when a term divides out from both the numerator and denominator). Examples: (a) Find the vertical asymptote for . ( )( ) 2xx2 2 4 (b) Find the vertical asymptote for fx . x2 9 x gx ( 2 )( ) x 25 3x2 12 (c) Find the vertical asymptote for hx . xx2 23 ( )( ) Pg 2 over Pre Calculus (2.6) Rational Functions & Asymptotes x 2 (d) Find the vertical asymptote for fx . x2 4 ( ) ( )( ) There is a hole at (2, ¼) and a vertical asymptote at x = – 2 sin x (e) Find the vertical asymptote for hx . x3 x (f) Find the vertical asymptote for hx . xx2 ( ) ( ) There is a hole at (0, – 1) and a vertical asymptote at x = 1 Pg 3 over Pre Calculus (2.6) Rational Functions & Asymptotes Definition of a Horizontal Asymptote nn1 Nx ann x a1 x a 1 x a 0 Let f be the rational function: fx mm1 D x bmm x b1 x b 1 x b 0 The line yb is a horizontal asymptote of the graph of a function f if approaches b as x increases or decreases without bound. Symbolically this means that, f() x b1 as x or f() x b2 as x . fx Strategies for finding Horizontal Asymptotes The degree of the numerator is n. The degree of the denominator is m. 1. If nm , the x-axis, or y 0, is the horizontal asymptote of the graph of f. 2. If nm , the line y anm b is the horizontal asymptote of the graph of f. 3. If nm , the graph of f has no horizontal asymptote. Examples: 3 (a) Find any horizontal asymptotes for cx . x 1 3x (b) Find any horizontal asymptotes for fx . x2 1 3x2 (c) Find any horizontal asymptotes for gx . x2 1 3x3 (d) Find any horizontal asymptotes for hx . x2 1 Pg 4 over Pre Calculus (2.6) Rational Functions & Asymptotes x 10 (e) Find any horizontal asymptotes for tx . x 2 ( ) a b c Table of values graph expanded graph c sin x (f) Find any horizontal asymptotes for tx . x3 Pg 5 Pre Calculus (2.6) Rational Functions & Asymptotes Applications Example: (a) A drug is administered to a patient and the concentration of the drug in the bloodstream is monitored. At time t 0 (in hours since giving the drug), the 5t concentration (in mg/L) is given by ct . Graph the function ct t 2 1 and include any asymptote(s). (1) What is the highest concentration of drug that is reached in the patient’s bloodstream? (2) What happens to the drug concentration after a long period of time? (3) How long does it take for the concentration to drop below 0.3 mg/L? (1) 2.5 mg/L (2) n=1 and m = 2, n<m so y = 0 therefore ( ) (3) √ 16.6 hours Pg 6 .