Math 20A Midterm Exam 1 (version 1) Solution
1. (6 points) Let
f
(
x
) =
e
2
x
and
g
(
x
) = 2 ln(
x
).
(a) Find the domain and range of
f
(
x
).
The domain of
f
is
R
,
the set of all real numbers and the range of
f
is
(0
,
∞
),
the set of all positive real
numbers.
(b) Does
f
(
x
)
have an inverse function? Justify your answer.
Yes,
f
(
x
)
has an inverse function because it is an increasing function, hence one-to-one.
(c) Are the functions
f
(
x
)
and
g
(
x
)
inverses of each other? Explain how you determined your
answer.
No:
g
(
x
) is not the inverse of
f
(
x
)
since
f
(
g
(
x
)) =
e
2
g
(
x
)
=
e
2
·
2ln(
x
)
=
x
4
negationslash
=
x
. (Note: it would be just as good
to observe that
g
(
f
(
x
))
negationslash
=
x
.)
2. (6 points) For each of the following limits, either evaluate it or explain why it does not exist.
(a) lim
x
→
0
braceleftbigg
1
2
x
-
1
x
(
x
+ 2)
bracerightbigg
lim
x
→
0
1
2
x
−
1
x
(
x
+ 2)
ff
= lim
x
→
0
(
x
+ 2)
−
2
2
x
(
x
+ 2)
= lim
x
→
0
x
2
x
(
x
+ 2)
= lim
x
→
0
1
2(
x
+ 2)
=
1
4
(b) lim
x
→
7
2
x
2
x
-
7
lim
x
→
7
2
x
2
x
−
7
does not exist because
lim
x
→
7
+
2
x
2
x
−
7
=
∞
.
(Note that
lim
x
→
This
preview
has intentionally blurred sections.
Sign up to view the full version.
This is the end of the preview.
Sign up
to
access the rest of the document.
- Fall '08
- staff
- Real Numbers, Derivative, lim, Continuous function
-
Click to edit the document details
