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3.4 Rational Functions I

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A rational function is a function of the form Where p and q are polynomial functions and q is not the zero polynomial. The domain consists of all real numbers except those for which the denominator is 0.

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Find the domain of the following rational functions: All real numbers except -6 and-2. All real numbers except -4 and 4. All real numbers.

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y = L y = R(x) y x y = L y = R(x) y x Horizontal Asymptotes

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x = c y x y x Vertical Asymptotes

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If an asymptote is neither horizontal nor vertical it is called oblique. y x

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Theorem Locating Vertical Asymptotes A rational function In lowest terms, will have a vertical asymptote x = r, if x - r is a factor of the denominator q.

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Vertical asymptotes: x = -1 and x = 1 No vertical asymptotes Vertical asymptote: x = -4 Find the vertical asymptotes, if any, of the graph of each rational function.

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1. If n < m, then y = 0 is a horizontal asymptote of the graph of R. 2. If n = m, then y = a n / b m is a horizontal asymptote of the graph of R. 3. If n = m + 1, then y = ax + b is an oblique asymptote of the graph of R. Found using long division. 4. If n > m + 1, the graph of R has neither a horizontal nor oblique asymptote. End behavior found using long division. Consider the rational function

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Horizontal asymptote: y = 0 Horizontal asymptote: y = 2/3 Find the horizontal and oblique asymptotes if any, of the graph of

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Oblique asymptote: y = x + 6

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