Granillo, Yvette – Homework 4 – Due: Sep 22 2005, 3:00 am – Inst: Edward Odell
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The due time is Central
time.
001
(part 1 of 1) 10 points
Functions
f
and
g
are defined on (

10
,
10)
by their respective graphs in
2
4
6
8

2

4

6

8
4
8

4

8
f
g
Find all values of
x
where the sum,
f
+
g
, of
f
and
g
is continuous, expressing your answer
in interval notation.
1.
(

10
,

4)
[
(

4
,
3)
[
(3
,
10)
2.
(

10
,

4)
[
(

4
,
10)
3.
(

10
,
3)
[
(3
,
10)
correct
4.
(

10
,

4]
[
[3
,
10)
5.
(

10
,
10)
Explanation:
Since
f
and
g
are piecewise linear, they are
continuous
individually
on (

10
,
10) except
at their ‘jumps’,
i.e.
, at
x
=

4 in the case of
f
and
x
=

4
,
3 in the case of
g
. But the sum
of continuous functions is again continuous, so
f
+
g
is certainly continuous on
(

10
,

4)
[
(

4
,
3)
[
(3
,
10)
.
The only question is what happens at
x
0
=

4
,
3. To do that we have to check if
lim
x
→
x
0

{
f
(
x
) +
g
(
x
)
}
=
f
(
x
0
) +
g
(
x
0
)
=
lim
x
→
x
0
+
{
f
(
x
) +
g
(
x
)
}
.
Now at
x
0
=

4,
lim
x
→ 
4

{
f
(
x
) +
g
(
x
)
}
=

2 =
f
(

4) +
g
(

4)
=
lim
x
→ 
4+
{
f
(
x
) +
g
(
x
)
}
,
while at
x
0
= 3,
lim
x
→
3

{
f
(
x
) +
g
(
x
)
}
=

7
6
=

5 =
lim
x
→
3+
{
f
(
x
) +
g
(
x
)
}
.
Thus,
f
+
g
is continuous at
x
=

4, but not
at
x
= 3.
Consequently, on (

10
,
10) the
sum
f
+
g
is continuous at all
x
in
(

10
,
3)
[
(3
,
10)
.
keywords: Stewart5e,
002
(part 1 of 1) 10 points
Below is the graph of a function
f
.
2
4
6

2

4

6
2
4
6
8

2

4