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There is no such real number since all powers of 2

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Unformatted text preview: h x the graph of k passes the Horizontal Line Test which means k is one to one and k −1 exists. Computing k −1 is left as an exercise. (−) −1 (+) 1 y = k (x) As the previous example illustrates, the graphs of general algebraic functions can have features we’ve seen before, like vertical and horizontal asymptotes, but they can occur in new and exciting 2x ways. For example, k (x) = √x2 −1 had two distinct horizontal asymptotes. You’ll recall that rational functions could have at most one horizontal asymptote. Also some new characteristics like ‘unusual steepness’8 and cusps9 can appear in the graphs of arbitrary algebraic functions. Our next example first demonstrates how we can use sign diagrams to solve nonlinear inequalities. (Don’t panic. The technique is very similar to the ones used in Chapters 2, 3 and 4.) We then check our answers graphically with a calculator and see some of the new graphical features of the functions in this extended family. Example 5.3.2. Solve the following inequalities. Check you...
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