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LESSON 2
Composition of Functions
Why?
Submersibles can descend several miles
below the surface of the ocean. You can
write a function
d
(
t
) that gives the depth
of the submersible after
t
minutes and
a function
p
(
d
) that gives the pressure
at depth
d
. The composition of the
functions
p
[
d
(
t
)] gives the pressure on
the submersible after
t
minutes.
Now
1
Perform compositions
of functions.
2
Apply compositions of
functions.
Then
You performed
arithmetic
operations on
functions.
1
Perform Compositions of Functions
You have already combined functions with
arithmetic operations. Another method used to combine functions is a composition of
functions. In a
composition of functions
, the results of one function are used to evaluate
a second function.
Key Concept
Composition of Functions
Words
Suppose
f
and
g
are functions such that
the range of
g
is a subset of the domain
of
f
. Then the composition function
f
◦
g
can be described by
[
f
◦
g
](
x
)
=
f
[
g
(
x
)].
Model
C07-002A-888482
x
g
(
x
)
[
f
◦
g
](
x
)
f
[
g
(
x
)]
domain
of
g
range of
g
domain of
f
range
of
f
Given
f
(
x
)
=
x
2
−
4 and
g
(
x
)
=
2
x
+
1, find each value.
a.
[
f
◦
g
](3)
[
f
◦
g
](3)
=
f
[
g
(3)]
Composition of functions
=
f
(7)
g
(3)
=
2(3)
+
1
=
7
=
45
f
(7)
=
7
2
−
4
=
45
So, [
f
◦
g
](3)
=
45.
b.
[
g
◦
f
](3)
[
g
◦
f
](3)
=
g
[
f
(3)]
Composition of functions
=
g
(5)
f
(3)
=
3
2
−
4
=
5
=
11
g
(5)
=
2(5)
+
1
=
11
So, [
g
◦
f
](3)
=
11.
Guided Practice
Given
f
(
x
)
=
3
x
−
6 and
g
(
x
)
=
x
3
+
1, find each value.
1A.
[
f
◦
g
](5)
1B.
[
g
◦
f
](5)
Example 1
Evaluate Compositions of Functions
New
Vocabulary
composition of
functions
Mathematical
Practices
4
Model with
mathematics.
7
Look for and make use
of structure.
The composition of
f
and
g
,
denoted by
f
◦
g
or
f
[
g
(
x
)], is
read
f of g
. To hear more
pronunciations of
expressions, log into your
eStudent Edition
. Ask your
teacher or a partner for
clarification as you need it.
Robert Schwemmer,
CINMS,
NOAA
322
|
Lesson 5-2