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x
f
(
x
)
g
(
x
)
1
2
3
4
6
10
5
If
f
(
x
)
=
x
+
2,
g
(
x
)
= −
4
x
+
3, and
h
(
x
)
=
x
2
−
2
x
+
1, find each value.
[
f
◦
(
g
◦
h
)](2)
60.
[
g
◦
(
h
◦
f
)](
−
4)
61.
[
h
◦
(
f
◦
g
)](5)
62.
MULTIPLE REPRESENTATIONS
You will explore [
f
◦
g
](
x
) and [
g
◦
f
](
x
) if
f
(
x
)
=
x
2
+
1
and
g
(
x
)
=
x
−
3.
a. Tabular
Make a table showing values for practices [
f
◦
g
](
x
) and [
g
◦
f
](
x
).
b. Graphical
Use a graphing calculator to graph [
f
◦
g
](
x
), and [
g
◦
f
](
x
) on the same
coordinate plane.
c. Verbal
Explain the relationship between [
f
◦
g
](
x
), and [
g
◦
f
](
x
).
63.
REASONING
Copy and complete the table. Use the
following clues and logical reasoning to help you.
•
f
(
x
) and
g
(
x
) are linear functions.
•
[
f
◦
g
](2)
=
6
•
[
g
◦
f
](3)
=
10
Given that
f
(
x
)
=
mx
+
d
and
g
(
x
)
=
ax
2
+
bx
+
c
, find each composition.
64.
(
f
◦
g
)(
x
)
65.
(
g
◦
f
)(
x
)
66.
(
f
◦
f
)(
x
)
67.
Suppose
f
(
x
)
=
x
p
and
g
(
x
)
=
x
q
, where
p
and
q
are positive integers. What can you
say about the power of (
f
◦
g
)(
x
) and (
g
◦
f
)(
x
)? Explain.
H.O.T. Problems
Use
H
igher-
O
rder
T
hinking Skills
68.
OPEN-ENDED
Write two functions
f
(
x
) and
g
(
x
) such that (
f
◦
g
)(4)
=
0.
69.
ERROR ANALYSIS
Denise and Keiko were asked to find [
f
◦
g
](
x
) given that
f
(
x
)
=
6
x
+
5 and
g
(
x
)
=
2
x
−
1. Is either of them correct? Explain your
reasoning. If neither student is correct, provide the correct answer.
Denise
Keiko
C05_037A_903990
70.
CHALLENGE
Given that
f
(
x
)
=
3
x
+
4, find [
f
◦
f
◦
f
](2).
71.
REASONING
State whether each statement is
sometimes
,
always
, or
never
true.
Explain.
a.
The domain of two functions
f
(
x
) and
g
(
x
) that are composed
g
[
f
(
x
)] is restricted
by the domain of
f
(
x
).
b.
The domain of two functions
f
(
x
) and
g
(
x
) that are composed
g
[
f
(
x
)] is restricted
by the domain of
g
(
x
).
72.
WRITING INMATH
In the real world, why would you ever perform a composition
of functions?
59
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