Graphics Reference
In-Depth Information
x
x
1, sketch the graphs of
f
39 If ABS(
x
)
and
f
(
x
)
ABS and
ABS
f
.
40
The composite rule
If thefunction
g
:
A
R is continuous at
x
a
and thefunction
f
:
g
(
A
)
R
is continuous at
g
(
a
), provethat
f
g
is continuous at
x
a
.
41 Useqn 40 to provethat, if a real function
f
is continuous at
x
a
,
then the function ABS
f
is also continuous at
a
, where ABS is
defined as in qn 39.
42 For any real function
f
, wedefinea function
f
on thesame
domain as
f
by
f
(
x
)
(
f
(
x
)
f
(
x
)
). If
f
is continuous at
a
, prove
that
f
is also continuous at
a
.
Sketch the graph of
f
x
when
f
(
x
)
1.
43 For any real function
f
, wedefinea function
f
on thesame
domain as
f
by
f
(
x
)
(
f
(
x
)
f
(
x
)
). If
f
is continuous at
a
,
provethat
f
is also continuous at
a
.
Sketch the graph of
f
x
when
f
(
x
)
1.
44 For any two real numbers
a
and
b
, wedefine
a
b
when
b
a
,
when
a
b
.
max(
a
,
b
)
Provethat max(
a
,
b
)
(
a
b
)
a
b
.
Provethat
f
(
x
)
max(
f
(
x
), 0).
45 If real functions
f
and
g
havethesamedomain
A
, and areboth
continuous at
x
a
A
, provethat thefunction max(
f
,
g
), defined
by max(
f
,
g
)(
x
)
max(
f
(
x
),
g
(
x
)) is also continuous at
x
a
.
46 If
f
(
x
)
x
and
g
(
x
)
x
, identify the function max(
f
,
g
).
47 Let
r
: R
0
R denote the reciprocal function
r
(
x
)
1/
x
, and let
f
(
x
)
1
x
. What is themaximum domain of definition of
f
r
and of
r
f
? Are both these functions continuous throughout their
domains? Use a computer or a graphic calculator to examine the
graphs of
f
,
r
f
and
f
r
.
48 Assuming that thesinefunction is continuous on R (see chapter 11),
determine the largest subset of R on which thefunction
f
of qn 19
is continuous.
49 Use a computer or a graphic calculator to examine the graph of
y
x
sin 1/
x
.