MAC 1147 Exam #2b

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Instructions: Do all scratch work on the test itself. Make sure your final answersare clearly labelled. Be sure to simplify all answers whenever possible. SHOW ALLWORK ON THIS EXAM IN ORDER TO RECEIVE FULL CREDIT!!!

No. Score

1 /6

2 /4

3 /14

4 /10

5 /4

6 /4

7 /10

8 /8

9 /14

10 /14

11 /15

12 /8

13 /13

14 /16

15 /6

16 /4

Total /150

(1) (a) For the given functions f and g, find the composite function (f ◦ g)(x). (2points)

f(x) = x2 + 4; g(x) = x2 + 5

(a) x4 + 8x2 + 21 (b) x4 + 10x2 + 29 (c) x4 + 29

(d) x4 + 21 (e) None of the above

(b) Find the domain of the composite function (f ◦ g)(x). (4 points - 2 pointsfor the answer and 2 points for the steps)

f(x) =3

x− 7; g(x) =

√x− 3

(a) (3, 7) ∪ (7,∞) (b) [3, 7) ∪ (7,∞) (c) [3, 52) ∪ (52,∞)

(d) [3, 7) ∪ (7, 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

(ii)

(a) Yes (b) No

(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) (b)

(c) (d)

(iii) Find the inverse of f . State the domain and range of f and f−1. (10 points)

f(x) =3x

x + 2

(4) Graph the function. (10 points)

f(x) = −ex+2 + 1

(5) (i) Change the exponential expression to an equivalent expression involving alogarithm. (2 points)

63 = x

(a) log6 3 = x (b) logx 6 = 3 (c) log3 6 = x

(d) log3 x = 6 (e) None of the above

(ii) Change the logarithmic expression to an equivalent expression involving anexponent. (2 points)

lnx = 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 pointsfor the steps)

f(x) = log3

(64− x2

)(a) (−64, 64) (b) (−8, 8) (c) [−8, 8]

(d) (−∞,−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 loga-rithms. Express powers as factors. (4 points - 2 points for the answer and 2points for the steps)

log8

(8x3) 5√

1 + 3x

(x− 6)7, x > 6

(a) 5 log8 x + 45

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 + 1

5log8 (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 theanswer and 2 points for the steps)

36 log99√x + log9

(36x6

)− log9 36

(a) log9 x136 (b) log9 x

154

(c) log9 x109 (d) log9 x

10

(9) (i) Solve the equation. (4 points - 2 points for the answer and 2 points for thesteps)

5x−4 = 2

(a) 4 + log2 5 (b) −4 + log2 5 (c) 4 + log5 2 (d) −4 + log5 2

(ii) Solve the equation. (10 points)

32x + 3x+1 − 4 = 0

(10) (i) Solve the equation. (4 points - 2 points for the answer and 2 points for thesteps)

6 + 4 lnx = 15

(a) ln(94

)(b) 9

4 ln 1(c) e

94 (d) e9

4

(ii) Solve the equation. (10 points)

log5 (x + 3) = 1− log5 (x− 1)

(11) (i) Write out the first four terms of the sequence. (3 points)

an = 6n + 5

(a) 11, 17, 23, 29 (b) 5, 11, 16, 21 (c) 6, 11, 16, 21

(d) 5, 11, 17, 23 (e) None of the above

(ii) Find the common difference of the previous sequence. (3 points)

(a) 5 (b) −5 (c) 6 (d) −6 (e) None of theabove

(iii) Write out the first four terms of the sequence. (3 points)

an =3n

2n+1

(a) 34, 98, 2716, 8132

(b) 32, 94, 27

8, 8116

(c) 12, 34, 98, 2716

(d) 32, 92, 27

2, 81

2(e) None of the above

(iv) Find the common ratio of the previous sequence. (3 points)

(a) 32

(b) 23

(c) 3 (d) 2 (e) None of theabove

(v) Write out the first four terms of the sequence. (3 points)

a1 = 2 an = n + an−1

(a) 1, 4, 2, 5 (b) 2, 0, 1, 2 (c) 2, 4,−1, 5

(d) 4, 7, 11, 16 (e) None of the above

(12) The given pattern continues. Write down the formula for the general nth termof the sequence suggested by the pattern. (8 points)

1

3,4

4,16

5,64

6,256

7,1024

8, . . .

(13) (i) Write out the sum. (3 points)

5∑k=0

1

2k+1

(a) 12

+ 132

(b) 1 + 12

+ 14

+ · · ·+ 132

(c) 12

+ 14

+ 18

+ · · ·+ 132

(d) 14

+ 18

+ 116

+ · · ·+ 164

(e) None of the above

(ii) Express the sum using summation notation. (10 points)

3

5− 6

6+

9

7− 12

8+ · · · − 36

16

(14) Find the sum of the series. (4 points each - 2 points for the answer and 2 pointsfor the steps)

(i)12∑k=1

(k2 + 2k − 4

)(a) 680 (b) 650 (c) 854 (d) 758 (e) None of the

above

(ii)14∑k=6

(2k3)

(a) 22, 500 (b) 22, 050 (c) 21, 600 (d) 10, 800 (e) None of theabove

(iii)10 + 12 + 14 + 16 + 18 + · · ·+ 68

(a) 1140 (b) 1170 (c) 1320 (d) 1425 (e) None of theabove

(iv)

2 +1

2+

1

8+

1

32+ · · ·

(a) 52

(b) 25

(c) 38

(d) 12

(e) None of theabove

(15) (i) Find the 48th term of the arithmetic sequence with initial term a = 1 andcommon difference d = 12. (3 points)

(a) 589 (b) 577 (c) 565 (d) 553 (e) None of theabove

(ii) Find the 19th term of the geometric sequence with initial term 12048

andcommon ratio −2. (3 points)

(a) −128 (b) 128 (c) −64 (d) 64 (e) None of theabove

(16) Expand the expression using the Binomial Theorem. (4 points - 2 points for theanswer and 2 points for the steps)

(3x + 2)5

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