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Unformatted text preview: er 3 culminated with
the Real Factorization Theorem, Theorem 3.16, which says that all polynomial functions with real
coeﬃcients can be thought of as products of linear and quadratic functions. Our next step was to
enlarge our ﬁeld2 of study to rational functions in Chapter 4. Being quotients of polynomials, we
can ultimately view this family of functions as being built up of linear and quadratic functions as
well. So in some sense, Chapters 2, 3, and 4 can be thought of as an exhaustive study of linear and
quadratic3 functions and their arithmetic combinations as described in Section 1.6. We now wish
to study other algebraic functions, such as f (x) = x and g (x) = x2/3 , and the purpose of the
ﬁrst two sections of this chapter is to see how these kinds of functions arise from polynomial and
rational functions. To that end, we ﬁrst study a new way to combine functions as deﬁned below.
Definition 5.1. Suppose f and g are two functions. The composite of g with f , denoted g ◦ f ,
is deﬁned by the formula (g...
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