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3 x 42 graphs of rational functions 421 259 exercises

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Unformatted text preview: rational functions. One of the standard tools we will use is the sign diagram which was first introduced in Section 2.4, and then revisited in Section 3.1. In those sections, we operated under the belief that a function couldn’t change its sign without its graph crossing through the x-axis. The major theorem we used to justify this belief was the Intermediate Value Theorem, Theorem 3.1. It turns out the Intermediate Value Theorem applies to all continuous functions,1 not just polynomials. Although rational functions are continuous on their domains,2 Theorem 4.1 tells us vertical asymptotes and holes occur at the values excluded from their domains. In other words, rational functions aren’t continuous at these excluded values which leaves open the possibility that the function could change sign without crossing through the x-axis. Consider the graph of y = h(x) from Example 4.1.1, 1 recorded below for convenience. We have added its x-intercept at 2 , 0 for the discussion that follows. Suppose we wish to construct a sign diagram for h(x). Recall that the intervals where h(x) > 0, or (+),...
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