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Unformatted text preview: f . From this perspective, we see g ◦ f as a two step process taking an input x
and ﬁrst applying the procedure f then applying the procedure g . Abstractly, we have
g f x f (x)
g (f (x)) g◦f In the expression g (f (x)), the function f is often called the ‘inside’ function while g is often called
the ‘outside’ function. There are two ways to go about evaluating composite functions - ‘inside
out’ and ‘outside in’ - depending on which function we replace with its formula ﬁrst. Both ways
are demonstrated in the following example.
Example 5.1.1. Let f (x) = x2 − 4x, g (x) = 2 − √ x + 3, and h(x) = indicated composite functions. State the domain of each.
1. (g ◦ f )(x) 2x
. Find and simplify the
x+1 5. (h ◦ h)(x) 2. (f ◦ g )(x) 6. (h ◦ (g ◦ f ))(x) 3. (g ◦ h)(x)
4. (h ◦ g )(x) 7. ((h ◦ g ) ◦ f )(x) Solution.
1. By deﬁnition, (g ◦ f )(x) = g (f (x)). We now illustrate the two ways to evaluate this.
• inside out : We insert the expression f (x) into g ﬁrst to get
(g ◦ f )(x)...
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