These two functions f(x) and g(x), when used in such a stepwise fashion, can be represented by a single,
composite function, h(x). The function h(x) is g(x) with the input specified as the output of f(x); h(x)=g(f(x)), or more compactly, .
Returning to our example, let's color-code the functions to help us see what's going on. Obviously, color coding isn't necessary on homework problems.
Notice that the construction of the composite function is not commutative; that is,
. We will see examples of this in the problems.
Conventionally, a composite function will be written in simplest form. This involves expanding parentheses and collecting terms; e.g.,
Problems:
Functions f(x) and g(x) are given. For each problem, construct two composite functions, . Evaluate each composite function for x=2. (Grading: 20 points for each problem, 5 points for each part.)
1.
2.
3.
4.
5.
--- end ---
x2
f(x)
2
4
2x+1
g(x)
9
h(x)(gf)(x)
=
o
(
)
(
)
2
2
2
2
g(x)2x1
g()2()12()1
f(x)x:
f(x)f(x)x
if x=
gf(x)2x1
gf(2)2(2)
2, then
as befo
8
re.
119
=
=
=+
=+
=+=+
++
=
==
o
o
(
)
(
)
gf(x)fg(x)
¹
oo
(
)
(
)
2
2
fg(x)2x14x4x1
=+=+=
o
(
)
(
)
gf(x)andfg(x)
oo
2
f(x)2x:g(x)x
==
f(x)x1:g(x)x2
=+=-
2
f(x)x1:g(x)x2x1
=+=++
1
f(x)3x:g(x)
x
==
2
1
f(x)x:g(x)x
x
==-
x
2
f(x)
x
y
2x+1
g(x)
x
y
x2
f(x)
x
y
2x+1
g(x)
x
y
x
2
f(x)
2
4
2x+1
g(x)
2
5
x2
f(x)
2
4
2x+1
g(x)
2
5
x
2
f(x)
2
4
2x+1
g(x)
9
gasimm
mat101_slp_5_instr.docx
0
