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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,
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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