(fog)(x)
f(g(x))
fo g
(also called
—
fo
g()
=
)
2
g(x
-
—
3
-
3)2
x — 3, find the composite
= (x
52
3)
and g(x)
g(f(x))
2}
=
/2
(—cc,
-
2].
x, find each function and its domain.
(d) g°g
g(/)
—
f(f)) =f(/)
‘
—
I
=
=
e .v
0. For f2
to be defined we must have
x
4. Thus we have 0
x 4, so the domain of
=
{xjx
2)
=
fof
O}
g(.v) =
1
ample 6 that, in general, fog
g of. Remember, the
iction g is applied first and then
f is applied second. In
hat first subtracts 3 and theii squares; of
g
is the function
Dts 3.
=
=
f(g) =fC
If f(s)
2
of all x in the domain of g such that
g(s) is in the domain
s defined whenever both g(x) and
f(g(x)) are defined. Fig
g in terms of machines.
)flS
f and g, the composite function
is defined by
-
‘osition (or composite) of and g and
f
is denoted by
g
F
1.
Homework Hints available in TEC
—
—
= x/(x
f((x +
3)10)
and h(s)
=
(x +
(x
+ I
3)10
3)10
+
f(g(x + 3))
= ,
10
x
f1.gçIi(x)))
+ 1), g(x)
h)(x) =f(g(h(x)))
f(s)
=
x + 3.
g
°
h.
=
cos
(
2
x + 9), find functions
f, g,
and
Then
(fog o
= x
+ 9
g(x)
=
cos x
f(x)
2
=
(c) v
(e) y
—6
=
=
9)]2
=
©
2f(x + 6)
fC)
[cos(s +
F(s)
(1)
(d) v
=
—f
©
+ 4)
h)C) =f(g(h(x))) =f(g(x + 9)) =f(cos(x + 9))
h(s)
SOLUTION Since F(s)
[cos(x + 9)]2, the formula for F says: First add 9, then
take the
cosine of the result, and finally square. So we let
= fo
EXAMPLE 9 Decomposing a function Given F(x)
Ii such that F
1. Suppose the graph off is given. Write equations for the graphs
that are obtained from the graph of I as follows.
(a) Shift 3 units upward.
(h) Shift 3 units downward.
(c) Shift 3 units to the right.
(d) Shift 3 units to the left.
(e) Reflect about the s-axis.
(f) Reflect about the v-axis.
(g) Stretch vertically by a factor of 3.
(h) Shrink vertically by a factor of 3.
2. Explain how each graph is obtained from the graph of
v = f (a).
(a) y = f(s) + 8
(b) y = f(x + 8)
(c) y = 8f(x)
(d) y = f(8x)
(e) y = —f(s)
I
(f) y = 8f(x)
3. The graph oy =fC) is given. Match each equation
with its
graph and give reasons for your choices.
(a) y = f(x 4)
(h) y = f(s) + 3
mercs
Ii if
(fo g
fog o
fl)lX)
So far we have used composition to build complicated functio
ns from simpler ones. But
in calculus it is often useful to be able to decompose a compli
cated function into simpler
ones, as in the following example.
SOLUTION
EXAMPLE 8 Find
g