Math SLP Homework

Discussion and examples:

A function can be thought of as a machine with an input x and an output y. Two examples are f(x): x2=y and g(x): 2x+1=y. Schematically,

Assume the input in each case is 2. Then

Now suppose we put the two functions together, so that the output of f(x) becomes the input of g(x), as follows:

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,

Notice that the construction of the composite function is not commutative; that is,

. To see this, let’s try our example “reversed;” i.e.,

Conventionally, a composite function will be written in simplest form. This involves expanding parentheses and collecting terms, as follows.

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. f(x)=2x : g(x)= x2 Answers:
2. f(x)= x+1 : g(x)= x-2
3. f(x)= x+1 : g(x)= x2+2x+1

4. f(x)= 3x : g(x)=

5. f(x)= x2 : g(x)=
x2
f(x)
2
4
2x+1
g(x)
9
h(x)(gf)(x)
=
o
(
)
(
)
2
2
gf(x)g(f(x))2(x)1
if x=2, then
gf(x)g(f(x))21415
as before.
==+
==+=+=
o
o
(
)
(
)
gf(x)fg(x)
¹
oo
(
)
(
)
(
)
(
)
2
2
22
fg(x)f(g(x))2×1
if x=2, then
fg(x)2×1(2(2)1)(41)25.
==+
=+=+=+=
o
o
(
)
(
)
2
2
fg(x)2x14x4x1
=+=+=
o
(
)
(
)
gf(x)andfg(x)
oo
(
)
(
)
(
)
(
)
2
2
gf(x)4x
gf(2)16
fg(x)2x
fg(2)8
=
=
=
=
o
o
o
o
1
x
1
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