Unformatted text preview: und in Section 1.8. In fact,
all of the transformations in that section can be viewed in terms of composing functions with linear functions. 5.1 Function Composition 287 when we write h ◦ g ◦ f . The second property can also be veriﬁed using Deﬁnition 5.1. Recall that
the function I (x) = x is called the identity function and was introduced in Exercise 15 in Section
2.1. If we compose the function I with a function f , then we have (I ◦ f )(x) = I (f (x)) = f (x),
and a similar computation shows (f ◦ I )(x) = f (x). This establishes that we have an identity
for function composition much in the same way the real number 1 is an identity for real number
multiplication. That is, just as for any real number x, 1 · x = x · 1 = x , we have for any function
f , I ◦ f = f ◦ I = f . We shall see the concept of an identity take on great signiﬁcance in the next
section. Out in the wild, function composition is often used to relate two quantities which may not
be directly related, but have a variable in common, as illustrated in our next example.
Example 5.1.2. The surface area S...
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