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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. (−)
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
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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