This preview shows pages 1–2. Sign up to view the full content.
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 onetoone.
(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
·
2 ln(
x
)
=
x
4
n
=
x
. (Note: it would be just as good
to observe that
g
(
f
(
x
))
n
=
x
.)
2. (6 points) For each of the following limits, either evaluate it or explain why it does not exist.
(a) lim
x
→
0
b
1
2
x

1
x
(
x
+ 2)
B
lim
x
→
0
1
2
x
−
1
x
(
x
+ 2)
f
= lim
x
→
0
(
x
+ 2)
−
2
2
x
(
x
+ 2)
= lim
x
→
0
x
2
x
(
x
+ 2)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 04/24/2008 for the course MATH 20A taught by Professor Staff during the Fall '08 term at UCSD.
 Fall '08
 staff
 Real Numbers

Click to edit the document details