COMPOSITE FUNCTIONS
A. Let f(x) = and g(x) =
2
x
x
x2+
and
f
D(,0)(0,)=โโ โโชg
D(,2)(2,)
=
โโ โ โ โโช
B. Find (f g)(x) f[g(x)]=๎
22(x2)
f[(g)x] xx
x2
+
==
+
The domain of the composite seems to be , but we must also
(,0)(0,)
โ
โโโช
consider the domain of g(x).
C. The actual domain of the composite function is the intersection of the
"apparent domain" of the result and the domain of the "inner function"
[
]
[
]
(,0)(0,) (,2)(2,)โโ โ โโ โ โ โโชโฉ โช
Consider the graphs of the two domains placed one above the other:
apparent domain
The domain of the composite is the overlap (intersection) of the two graphs,
i.e.
[
]
[
]
(,0)(0,) (,2)(2,)(,2)(2,0)(0,)โโ โ โโ โ โ โ = โโ โ โ โโชโฉ โช โชโช
D. For Practice: Find each of the following compositions and their domains.
1. (g f)(x)๎
2. (f f)(x)๎
3. (g g)(x)๎
1. ; its domain is
1
(g f)(x) 1x
=+
๎(,1)(1,0)(0,)
โ
โโ โ โโชโช
2. ; its domain is
(f f)(x) x=๎(,0)(0,)
โ
โโโช
3. = ; its domain is
(g g)(x)๎x
3x 4+
44
(,2) 2, ,
33
โโโโ
โ
โโ โ โ โ โ
โโโโ
โโ โโ
โชโช
0
2โg
D