Adding and subtracting functions
The function f  g and its graph
.
Suppose that f and g are two functions defined on the interval
[
]
,
a b
such that
( )
f
x
>
( )
g
x
for
≤
a
x
≤
b
.
Then we can define a function h on the interval
[
]
,
a b
by
=
( )
h
x

( )
f
x
( )
g
x
for all numbers
x
with
≤
a
x
≤
b
.
At a given number
c
we can visualize the value of h as the distance between the point (
)
,
c
( )
g
c
on the graph of g and the point (
)
,
c
( )
f
c
vertically above it as indicated in the following picture.
The function h is denoted by
f

g
so that
(f

g)
=
( )
x

( )
f
x
( )
g
x
for all numbers
x
such that
≤
a
x
≤
b
.
Because of the condtion that
( )
f
x
>
( )
g
x
for
≤
a
x
≤
b
, the graph of the function f

g is above the
x
axis for
≤
a
x
≤
b
.
Given
any
two real valued functions f and g, we can define a function f

g
by the formula:
(f

g)
=
( )
x

( )
f
x
( )
g
x
.
In order for

( )
f
x
( )
g
x
to be defined for a real number
x
it is necessary that both
( )
f
x
and
( )
g
x
are defined.
It follows that the domain of
the function f

g must consist of real numbers that are
common
to the domains of both of the functions f and g, that is, it is the
intersection
of the domains of f and g.
We have
D(f

g)
=
D(f)
∩
D(g).
Example 1
:
Let f and g be the two functions defined by
=
( )
f
x

4
x
2
and
=
( )
g
x
3
x
respectively.
(a) State the domains of f, g and f

g.
(b) Sketch the graph of f, g and f

g.
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View Full DocumentSolution
:
(a)
The domain of f is the interval
{
x

≤

2
x
≤
2 }
=
[
]
,

2 2 , while the domain of g is the interval
{
x

x
>
0 }
=
[ ,
0
∞
).
The intersection of these two domains is the interval
{
x

≤
0
x
≤
2 }
=
[
]
,
0 2 ,
which is the domain of f

g.
(b) The graph of f is the part of the circle with its centre at the origin and radius 2 which lies on or above the
x
axis.
Since
=
3
x
3
x
for all real numbers
x
with
x
>
0, the graph of g is obtained by stretching the graph of
=
y
x
parallel to the
y
axis
with a scale factor of
3.
In the following picture, the graph of
=
y

4
x
2
, that is, the graph of f, is drawn in
purple
, while the graph of
=
y
3
x
, that is, the
graph of g, is drawn in
green
.
It looks as though the two graphs intersect where
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 Spring '08
 Uri
 Complex number, Peter Stone

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