MAC 1147 Exam #2b Answer Key
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Answer Key
Summer 2012
HONOR CODE: On my honor, I have neither given nor received any aid on this
examination.
Signature:
Instructions: Do all scratch work on the test itself. Make sure your final answers
are clearly labelled. Be sure to simplify all answers whenever possible. SHOW ALL
WORK ON THIS EXAM IN ORDER TO RECEIVE FULL CREDIT!!!
No.
1
2
3
4
5
6
7
8
Score
/6
/4
/14
/10
/4
/4
/10
/8
9
10
11
12
13
14
15
16
Total
/14
/14
/15
/8
/13
/16
/6
/4
/150
(1) (a) For the given functions f and g, find the composite function (f ◦ g)(x). (2
points)
f (x) = x2 + 4; g(x) = x2 + 5
(a) x4 + 8x2 + 21
(b) x4 + 10x2 + 29
(d) x4 + 21
(c) x4 + 29
(e) None of the above
(b) Find the domain of the composite function (f ◦ g)(x). (4 points - 2 points
for the answer and 2 points for the steps)
f (x) =
(a) (3, 7) ∪ (7, ∞)
√
3
; g(x) = x − 3
x−7
(b) [3, 7) ∪ (7, ∞)
(d) [3, 7) ∪ (7, 52) ∪ (52, ∞)
(c) [3, 52) ∪ (52, ∞)
(e) None of the above
(2) Determine whether or not each function is one-to-one. (2 points each)
(i) {(8, −2), (2, −8), (3, 5), (−3, −5)}
(a) Yes
(b) No
(a) Yes
(b) No
(ii)
(3)
(i) Given the function f , determine its inverse. (2 points)
{(−3, 4), (−1, 5), (0, 2), (2, 4), (5, 7)}
(a) {(−3, −4), (−1, −5), (0, −2), (2, −4), (5, −7)}
(b) {(4, −3), (5, −1), (2, 0), (4, 2), (7, 5)}
(c) {(3, −4), (1, −5), (0, −2), (−2, −4), (−5, −7)}
(d) {(3, 4), (1, 5), (0, 2), (−2, 4), (−5, 7)}
(ii) Given the graph of f , determine what the graph of f −1 looks like. (2 points)
(a)
(c)
(b)
(d)
(iii) Find the inverse of f . State the domain and range of f and f −1 . (10 points)
f (x) =
3x
x+2
2x
2x
=−
3−x
x−3
Domain of f = (−∞, −2) ∪ (−2, ∞)
f −1 (x) =
Range of f = (−∞, 3) ∪ (3, ∞)
Domain of f −1 = (−∞, 3) ∪ (3, ∞)
Range of f −1 = (−∞, −2) ∪ (−2, ∞)
(4) Graph the function. (10 points)
f (x) = −ex+2 + 1
(5)
(i) Change the exponential expression to an equivalent expression involving a
logarithm. (2 points)
63 = x
(a) log6 3 = x
(b) logx 6 = 3
(d) log3 x = 6
(c) log3 6 = x
(e) None of the above
(ii) Change the logarithmic expression to an equivalent expression involving an
exponent. (2 points)
ln x = 7
(a) x7 = e
(b) ex = 7
(c) e7 = x
(d) 7e = x
(6) Find the domain of the function. (4 points - 2 points for the answer and 2 points
for the steps)
f (x) = log3 64 − x2
(a) (−64, 64)
(d) (−∞, −8) ∪ (8, ∞)
(b) (−8, 8)
(c) [−8, 8]
(e) None of the above
(7) Graph the function. (10 points)
f (x) = − log4 (x − 1) + 3
(8)
(i) Write the single logarithmic expressions as a sum and/or difference of logarithms. Express powers as factors. (4 points - 2 points for the answer and 2
points for the steps)
√
(8x3 ) 5 1 + 3x
log8
, x>6
(x − 6)7
(a) 5 log8 x + 54 log8 (1 + 3x) − 7 log8 (x − 6)
(b) 1 + 3 log8 x − 5 log8 (1 + 3x) − 7 log8 (x − 6)
(c) log8 5 + log8 x3 + 51 log8 (1 + 3x) − log8 (x − 6) − log8 7
(d) 1 + 3 log8 x + 15 log8 (1 + 3x) − 7 log8 (x − 6)
(ii) Express as a single logarithm in simplest terms. (4 points - 2 points for the
answer and 2 points for the steps)
√
36 log9 9 x + log9 36x6 − log9 36
13
(a) log9 x 6
10
(c) log9 x 9
15
(b) log9 x 4
(d) log9 x10
(9)
(i) Solve the equation. (4 points - 2 points for the answer and 2 points for the
steps)
5x−4 = 2
(a) 4 + log2 5
(b) −4 + log2 5 (c) 4 + log5 2
(ii) Solve the equation. (10 points)
32x + 3x+1 − 4 = 0
x = 0 (x = log3 (−4) not allowed)
(d) −4 + log5 2
(10)
(i) Solve the equation. (4 points - 2 points for the answer and 2 points for the
steps)
6 + 4 ln x = 15
9
9
(a) ln 49
(b) 4 ln9 1
(d) e4
(c) e 4
(ii) Solve the equation. (10 points)
log5 (x + 3) = 1 − log5 (x − 1)
x = 2 (x = −4 not allowed)
(11)
(i) Write out the first four terms of the sequence. (3 points)
an = 6n + 5
(b) 5, 11, 16, 21
(a) 11, 17, 23, 29
(d) 5, 11, 17, 23
(c) 6, 11, 16, 21
(e) None of the above
(ii) Find the common difference of the previous sequence. (3 points)
(a) 5
(b) −5
(d) −6
(c) 6
(e) None of the
above
(iii) Write out the first four terms of the sequence. (3 points)
an =
(a)
(d)
(b)
3 9 27 81
, , ,
4 8 16 32
3 9 27 81
, , ,
2 4 8 16
3 9 27 81
, , ,
2 2 2 2
3n
2n+1
(c)
1 3 9 27
, , ,
2 4 8 16
(e) None of the above
(iv) Find the common ratio of the previous sequence. (3 points)
(a)
3
2
(b)
2
3
(c) 3
(d) 2
(e) None of the
above
(v) Write out the first four terms of the sequence. (3 points)
a1 = 2 an = n + an−1
(a) 1, 4, 2, 5
(d) 4, 7, 11, 16
(b) 2, 0, 1, 2
(c) 2, 4, −1, 5
(e) None of the above
(12) The given pattern continues. Write down the formula for the general nth term
of the sequence suggested by the pattern. (8 points)
1 4 16 64 256 1024
, , , ,
,
,...
3 4 5 6 7
8
an =
4n−1
n+2
(13)
(i) Write out the sum. (3 points)
5
X
1
2k+1
k=0
1
32
(a)
1
2
+
(d)
1
4
+ 81 +
(b) 1 + 21 + 14 + · · · +
1
16
+ ··· +
1
64
1
32
(c)
1
2
+ 14 + 18 + · · · +
(e) None of the above
(ii) Express the sum using summation notation. (10 points)
3 6 9 12
36
− + −
+ ··· −
5 6 7
8
16
12
X
k=1
(−1)
k+1
12
X
3n
3k
or
(−1)n−1
k+4
n+4
n=1
(other answers possible)
1
32
(14) Find the sum of the series. (4 points each - 2 points for the answer and 2 points
for the steps)
(i)
12
X
k 2 + 2k − 4
k=1
(a) 680
(b) 650
(c) 854
(d) 758
(e) None of the
above
(ii)
14
X
2k 3
k=6
(a) 22, 500
(b) 22, 050
(c) 21, 600
(d) 10, 800
(e) None of the
above
(iii)
10 + 12 + 14 + 16 + 18 + · · · + 68
(a) 1140
(b) 1170
(c) 1320
(d) 1425
(e) None of the
above
(iv)
2+
(a)
5
2
(b)
2
5
1 1
1
+ +
+ ···
2 8 32
(c)
3
8
(d)
1
2
(e)
None of the
above
(15)
(i) Find the 48th term of the arithmetic sequence with initial term a = 1 and
common difference d = 12. (3 points)
(a) 589
(b) 577
(c) 565
(d) 553
(e) None of the
above
(ii) Find the 19th term of the geometric sequence with initial term
common ratio −2. (3 points)
(a) −128
(b) 128
(c) −64
(d) 64
1
2048
and
(e) None of the
above
(16) Expand the expression using the Binomial Theorem. (4 points - 2 points for the
answer and 2 points for the steps)
(3x + 2)5
243x5 + 810x4 + 1080x3 + 720x2 + 240x + 32
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