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pre-calc 2.6 & 2.7 - Section 2.6 Rational Functions...

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Section 2.6 Rational Functions and Their Graphs Rational Functions are quotients of polynomial function. ) ( ) ( ) ( x q x p x f = , where p and q are polynomials with 0 ) ( x q . domain of rational function – set of all real numbers except the x-values that make the denominator zero EX 1: Find the domain of each rational function. a.) 5 25 ) ( 2 - - = x x x f b.) 25 ) ( 2 - = x x x h c.) 16 2 ) ( 2 + = x x x f Reciprocal Function : x x f 1 ) ( = Another Basic rational function: 2 1 ) ( x x f = Arrow notation symbol meaning + a x x approaches a from the right - a x x approaches a from the left x x approaches infinity; that is, x increases without bound -∞ x x approaches negative infinity; that is, x decreases without bound In calculus, you will use limits to convey ideas involving a function’s end behavior or its possible asymptotic behavior. example: 0 1 lim = x x EX 2:
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The rules for transformations are the same as earlier. *Vertical Asymptote - The line x = a is a vertical asymptote of the graph of f if f(x) increases or decreases without bound as x approaches a. -- Locating vertical asymptotes : If ) ( ) ( ) ( x q x p x f = is a rational function in which p(x) and q(x) have no common factors and a is zero of q(x) , then x = a is a vertical asymptote of the graph of f . EX 3: Find the vertical asymptotes, if any, of the graph of each rational function: a.) 1 ) ( 2 - = x x x f b.) 36 6 ) ( 2 - - = x x x g c.) 25 5 ) ( 2 + - = x x x h Note : If a is a root of p(x) and q(x)
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