Graphing6 g and g 1 on the same set of axes shows

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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? 5 A more mathematical example in which the order of two processes matters can be fo...
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