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**Unformatted text preview: **= g (f (x)) = g x2 − 4x = 2 −
Hence, (g ◦ f )(x) = 2 − √ x2 − 4x + 3. (x2 − 4x) + 3 = 2 − x2 − 4x + 3 5.1 Function Composition 281 • outside in : We use the formula for g ﬁrst to get
(x2 − 4x) + 3 = 2 −
√
We get the same answer as before, (g ◦ f )(x) = 2 − x2 − 4x + 3.
(g ◦ f )(x) = g (f (x)) = 2 − f (x) + 3 = 2 − x2 − 4x + 3 To ﬁnd the domain of g ◦ f , we need to ﬁnd the elements in the domain of f whose outputs
f (x) are in the domain of g . We accomplish this by following the rule set forth in Section
1.5, that is, we ﬁnd the domain before we simplify. To that end, we examine (g ◦ f )(x) =
2 − (x2 − 4x) + 3. To keep the square root happy, we solve the inequality x2 − 4x + 3 ≥ 0
by creating a sign diagram. If we let r(x) = x2 − 4x + 3, we ﬁnd the zeros of r to be x = 1
and x = 3. We obtain
(+) 0 (−) 0 (+)
1 3 Our solution to x2 − 4x + 3 ≥ 0, and hence the domain of g ◦ f , is (−∞, 1] ∪ [3, ∞).
2. To ﬁnd (f ◦ g )(x), we ﬁnd f (g (x)).
• in...

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