Though not absolutely necessary4 it is good practice

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Unformatted text preview: f Vertical Asymptotes and Holes:a Suppose r is a rational function ( x) which can be written as r(x) = p(x) where p and q have no common zeros.b Let c be a real number q which is not in the domain of r. (c) • If q (c) = 0, then the graph of y = r(x) has a hole at c, p(c) . q • If q (c) = 0, then the the line x = c is a vertical asymptote of the graph of y = r(x). a b Or, ‘How to tell your asymptote from a hole in the graph.’ In other words, r(x) is in lowest terms. In English, Theorem 4.1 says if x = c is not in the domain of r but, when we simplify r(x), it no longer makes the denominator 0, then we have a hole at x = c. Otherwise, we have a vertical asymptote. Example 4.1.2. Find the vertical asymptotes of, and/or holes in, the graphs of the following rational functions. Verify your answers using a graphing calculator. 1. f (x) = 2x 2−3 x 3. h(x) = x2 − x − 6 x2 + 9 2. g (x) = x2 − x − 6 x2 − 9 4. r(x) = x2 − x − 6 x2 + 4 x + 4 Solution. 1. To use Theorem 4.1, we first find all of the √ numbers which aren’t in the domain of f . To real 2 − 3 = 0 and get x = ± 3. Since the expression f (x) is in lowest terms, do so, we solve x √ √ there is no...
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