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Unformatted text preview: (h ◦ g ) ◦ f , provided the composite functions are deﬁned.
• If I is deﬁned as I (x) = x for all real numbers x, then I ◦ f = f ◦ I = f .
By repeated applications of Deﬁnition 5.1, we ﬁnd (h ◦ (g f ))(x) = h((g ◦ f )(x)) = h(g (f (x))).
Similarly, ((h ◦ g ) ◦ f )(x) = (h ◦ g )(f (x)) = h(g (f (x))). This establishes that the formulas for the
two functions are the same. We leave it to the reader to think about why the domains of these
two functions are identical, too. These two facts establish the equality h ◦ (g ◦ f ) = (h ◦ g ) ◦ f .
A consequence of the associativity of function composition is that there is no need for parentheses
4 This shows us function composition isn’t commutative. An example of an operation we perform on two functions
which is commutative is function addition, which we deﬁned in Section 1.6. In other words, the functions f + g and
g + f are always equal. Which of the remaining operations on functions we have discussed are commutative?
A more mathematical example in which the order of two processes matters can be fo...
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