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The composition of two functions may not exist. Given two functions
f
and
g
,
[
f
◦
g
](
x
) is defined only if the range of
g
(
x
) is a subset of the domain of
f
. Likewise,
[
g
◦
f
](
x
) is defined only if the range of
f
(
x
) is a subset of the domain of
g
.
For each pair of functions, find [
f
◦
g
](
x
) and [
g
◦
f
](
x
), if they exist. State the
domain and range for each composed function.
a.
f
=
{(1, 8), (0, 13), (15, 11), (14, 9)},
g
=
{(8, 15), (5, 1), (10, 14), (9, 0)}
To find
f
◦
g
, evaluate
g
(
x
) first. Then use the range to evaluate
f
(
x
).
f
[
g
(8)]
=
f
(15) or 11
g
(8)
=
15
f
[
g
(5)]
=
f
(1) or 8
g
(5)
=
1
f
[
g
(10)]
=
f
(14) or 9
g
(10)
=
14
f
[
g
(9)]
=
f
(0) or 13
g
(9)
=
0
f
◦
g
=
{(8, 11), (5, 8), (10, 9), (9, 13)}
D
=
{5, 8, 9, 10}, R
=
{8, 9, 11, 13}
To find
g
◦
f
, evaluate
f
(
x
) first. Then use the range to evaluate
g
(
x
).
g
[
f
(1)]
=
g
(8) or 15
f
(1)
=
8
g
[
f
(0)]
=
g
(13)
g
(13) is undefined.
g
[
f
(15)]
=
g
(11)
g
(11) is undefined.
g
[
f
(14)]
=
g
(9) or 0
f
(14)
=
0
Because 11 and 13 are not in the domain of
g
,
g
◦
f
is undefined for
x
=
0 and
x
=
15. However,
g
[
f
(1)]
=
15 and
g
[
f
(14)]
=
0.
So,
g
◦
f
=
{(1, 15), (14, 0)}.
D
=
{1, 14}, R
=
{0, 15}
b.
f
(
x
)
=
2
x
−
5,
g
(
x
)
=
4
x
[
f
◦
g
](
x
)
=
f
[
g
(
x
)]
Composition of functions
[
g
◦
f
](
x
)
=
g
[
f
(
x
)]
=
f
(4
x
)
Substitute.
=
g
(2
x
−
5)
=
2(4
x
)
−
5
Substitute again.
=
4(2
x
−
5)
=
8
x
−
5
Simplify.
=
8
x
−
20
So, [
f
◦
g
](
x
)
=
8
x
−
5 and [
g
◦
f
](
x
)
=
8
x
−
20.
For [
f
◦
g
](
x
), D
=
{all real numbers} and R
=
{all real numbers}, and for [
g
◦
f
](
x
),
D
=
{all real numbers} and R
=
{all real numbers}.
Guided Practice
2A.
f
(
x
)
=
{(3,
−
2), (
−
1,
−
5), (4, 7), (10, 8)},
g
(
x
)
=
{(4, 3), (2,
−
1), (9, 4), (3, 10)}
2B.
f
(
x
)
=
x
2
+
2 and
g
(
x
)
=
x
−
6
Example 2
Perform Compositions of Functions
Notice that in most cases,
f
◦
g
≠
g
◦
f
. Therefore, the order in which two functions are
composed is important.
Study Tip
Composition
Be careful not
to confuse a composition
f
[
g
(
x
)] with multiplication
of functions (
f
·
g
)(
x
).
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