3_7 - q • For each real zero r x = r is the vertical...

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Section 3.7: Rational Functions Instructor: Ms. Hoa Nguyen ([email protected]) 1 Rational Function A rational function f is a function of the form f ( x ) = p ( x ) q ( x ) where p ( x ) = a n x n + a n - 1 x n - 1 + ··· + a 1 x + a 0 ( a n 6 = 0) and q ( x ) = b m x m + b m - 1 x m - 1 + ··· + b 1 x + b 0 ( b m 6 = 0). The domain of a rational function f is the set of all x such that q ( x ) 6 = 0. Example 1 Example 2 1
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2 Horizontal Asymptote The line of y = L is a horizontal asymptote of the graph of y = f ( x ) if f ( x ) L as x → ∞ or f ( x ) L as x → -∞ . The graph of f ( x ) = 1 ( x - 2) 2 . Finding horizontal asymptotes : Let f ( x ) = p ( x ) q ( x ) . The degree of p ( x ) is n and the degree of q ( x ) is m . If n < m (i.e., f is proper), y = 0 is the horizontal asymptote. If n = m , y = a n b m is the horizontal asymptote. Otherwise, NO horizontal asymptotes. Example 3 2
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3 Vertical Asymptote The line of x = c is a vertical asymptote of the graph of y = f ( x ) if | f ( x ) | → ∞ as x c . The graph of f ( x ) = x +1 x - 1 . Note: If p ( x ) and q ( x ) of a rational function f ( x ) = p ( x ) q ( x ) have NO common factors, then f is said to be in the lowest term . Finding vertical asymptotes : Let f ( x ) = p ( x ) q ( x ) in the LOWEST term . Let q ( x ) = 0 to find the real zeros of
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Unformatted text preview: q . • For each real zero r , x = r is the vertical symptote. Remark : If f is NOT in the LOWEST term, following the above steps may result in an incorrect listing of vertical asymptotes. Example 4 3 4 Oblique Asymptote The graph of f ( x ) = x 3 +2 x 2-1 x 2 + x +1 . Question : Based on the result of the long division, what line does f approach as | x | → ∞ ? Finding oblique asymptotes : ONLY when n = m + 1, there is an oblique asymptote which is the QUOTIENT computed in the long division. Otherwise, there is NO oblique asymptote. Example 5 5 Inverse Variation The number y varies inversely with the number x if y = k x for some k 6 = 0. In other words, y = f ( x ) where f ( x ) = k x for some k 6 = 0. 4 LEFT: f ( x ) = k x where k > 0. RIGHT: f ( x ) = k x where k < 0. 5...
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This note was uploaded on 05/23/2011 for the course MAC 1147 taught by Professor Nuegyen during the Spring '11 term at FSU.

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3_7 - q • For each real zero r x = r is the vertical...

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