mac1147_lecture12_1_b

mac1147_lecture12_1_b - L12 Rational Functions: Asymptotes...

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149 L12 Rational Functions: Asymptotes and Holes; Graphing Rational Function; Inequalities A rational function is a function of the form () p x Rx qx = where p and q are polynomials and q is not the zero polynomial. The domain of the rational function ( ) R x is The zeros of the rational function ( ) R x are Example : Find the domain and zeros of the function 2 2 (2 ) xx = Note : Always give the domain of the original fraction, not the one reduced to lowest terms .

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150 Vertical Asymptotes Example : Consider the function 2 2 6( 3 ) ( 2 ) 2 () 23 (3 ) (1 ) 1 x xx x x Rx x +− + == = + , 3 x ≠ − Describe its behavior as 1 x . A line x a = is a vertical asymptote of the graph of R x if →∞ as x a or x a + . Vertical asymptotes occur at real zeros of the denominator of the reduced to lowest terms rational function. Note : The graph of a rational function never intersects a vertical asymptote. (Why?)
151 Holes The graph of ( ) R x has a hole (removable discontinuity) at a real number x a = if ( ) R x is not defined at a , but the factor ( ) x a can be canceled out of the denominator (removed from the denominator). In order to find the y -coordinate of the hole, plug in x a = into the reduced expression for ( ) R x . Example: Find the vertical asymptotes and holes, if any. 23 () 1 x fx x = 2 1 34 x gx xx + = −−

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152 Oblique and Horizontal Asymptotes Example : Consider the function () 1 1 fx x = + .
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This note was uploaded on 11/07/2011 for the course MAC 1147 taught by Professor German during the Summer '08 term at University of Florida.

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mac1147_lecture12_1_b - L12 Rational Functions: Asymptotes...

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