Stitz-Zeager_College_Algebra_e-book

To check this analytically we rst check g 1 g x x for

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Unformatted text preview: x2 − 4x) + 3 (x2 − 4x) + 3 √ 4 − 2 x2 − 4x + 3 √ 3 − x2 − 4x + 3 3− • outside in : We use the formula for (h ◦ g )(x) first to get ((h ◦ g ) ◦ f )(x) = (h ◦ g )(f (x)) 286 Further Topics in Functions = = = 4−2 (f (x)) + 3 3− f (x)) + 3 4−2 (x2 − 4x) + 3 (x2 − 4x) + 3 √ 4 − 2 x2 − 4x + 3 √ 3 − x2 − 4x + 3 3− We note that the formula for ((h ◦ g ) ◦ f )(x) before simplification is identical to that of (h ◦ (g ◦ f ))(x) before √ simplified it. Hence, the two functions have the same domain, we √ √ √ h ◦ (f ◦ g ) is (−∞, 2 − 10) ∪ (2 − 10, 1] ∪ 3, 2 + 10 ∪ 2 + 10, ∞ . It should be clear from Example 5.1.1 that, in general, when you compose two functions, such as f and g above, the order matters.4 We found that the functions f ◦ g and g ◦ f were different as were g ◦ h and h ◦ g . Thinking of functions as processes, this isn’t all that surprising. If we think of one process as putting on our socks, and the other as putting on our shoes, the order in which we do these two tasks does matter.5 Also note the importan...
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