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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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