Multiplying and dividing functions
The functions f . g and f/g
.
Let f and g be
any
two real valued functions. We can define a function
f . g on a domain which consists of real numbers that are
common
to the domains of both of the functions f and g, that is, on the intersection of the domains of f and g, by the formula:
(f . g)
=
(
)
x
(
)
f
x
.
(
)
g
x
.
We have
D(f . g)
=
D(f)
∩
D(g).
We can also define a function f / g by the formula:
(f / g)
=
(
)
x
(
)
f
x
/
(
)
g
x
.
Since division by zero must be avoided, the domain of f / g consists of all real numbers
x
such that both (
)
f
x
and (
)
g
x
are defined and such
that
≠
(
)
g
x
0. Thus
D(f / g)
=
{ x

∈
x
D(f)
and
∈
x
D(g)
and
≠
(
)
g
x
0 }
=
{
x

∈
x
D(f)
∩
D(g)
and
≠
(
)
g
x
0
}.
Example 1
:
Let f and g be the two functions defined by
=
(
)
f
x
+
4
x
and
=
(
)
g
x

4
x
respectively.
(a)
State the domains of f, g,
f
.
g and f / g.
(b)
Give simplified formulas to describe the functions
f
.
g and f / g, and sketch the graphs of these two functions.
Solution
:
(a) The domain of f is the set of all real numbers that are greater than or equal to

4, that is,
D(f)
=
{
x

x
>
4 }
=
[
,

4
∞
).
The domain of g is the set of all real numbers that are less than or equal to 4, that is,
D(g)
=
{
x

≤
x
4 }
=
(
,
∞
4 ].
We have
D(f
.
g)
=
{
x

x
>

4
and
≤
x
4 }
=
{
x

≤

4
x
≤
4 }
=
[
]
,

4 4 .
Since
=
(
)
g
x
0
when
=
x
4, we have
D( f / g )
=
{
x

≤

4
x
<
4 }
=
[
,

4
4 ).
(b)
( f
.
g )
=
(
)
x
+
4
x .

4
x
=
(
)
+
4
x
(
)

4
x
=

16
x
2
.
The graph of
f
.
g
is the part of the circle with centre at the origin and radius 4 that lies on or above the
x
axis.
( f / g )
=
(
)
x
+
4
x

4
x
=
+
4
x

4
x
.
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We can obtain an alternative expression for
+
4
x

4
x
by performing the following polynomial division.
1
______

+
x
4

+
x
4

x
4
_____
8
This division shows that
=
+
4
x

4
x

+
1
8

4
x
=


8

x
4
1.
The graph of
=
y


8

x
4
1 is a rectangular hyperbola obtained by applying appropriate transformations to the graph of
=
y
1
x
.
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 Spring '08
 Uri
 Real Numbers, Continuous function, Complex number, Peter Stone

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