FALL 2006, MATH 11011-COLLEGE ALGEBRA SOLUTIONS TO TEST 4
PART I. (Multiple Choice–Short Answer)
1. Which of the graphs below belong to 1-1 functions? (For full credit mark all answers that fit.) (b) and (d)
2. Which of the following functions are one-to-one? (For full credit mark all answers that fit.)
(a) f (x) = |x + 2| − 3
(b) h(x) = −(x − 2)2 + 1
(c) g(x) = 3x − 1
√
(d) k(x) = − −x − 2 + 3
Correct Answer
Correct Answer
3. Which of the following graphs represents the inverse of the graph of the 1-1 function f ?
The answer is (d)
4. For large values of n the quantity (1 + n1 )n approaches...
(a) 1
(b) 0
(c) e
(d)
1
e
After all, (1 + nx )n → ex as x → ∞.
5. Solve for x: log3 x = 4
(a) 12
log3 x = 4 ←→ x = 34 = 81
(b) 64
(c) 4/3
(d) 81
6. The domain of f (x) = log7 x is...
(a) (0, ∞)
(c) [0, ∞)
(b) All reals except zero
(d) All reals
7. If you invest money at an annual rate of 8% compounded continuously, how long will it take for your investment
to triple ?
(a) 8 years
(b)
ln 3
0.08
(c) 8 ln 3 years
years
Solve P e0.08t = 3P : e0.08t = 3 → 0.08t = ln 3 → t =
ln 3
0.08
(d) It would never really triple
years.
8. Which of the following intervals contains a solution of the equation 35x−2 = 27x+1 ?
(a) (5, ∞)
(b) (−∞, 2]
(c) (−8, 3]
35x−2 = 27x+1 → 35x−2 = 33(x+1) → 5x − 2 = 3x + 3 → x =
(d) There are no solutions
5
2
9. Which of the following intervals contains a solution of the equation log(x − 3) + log x = 1 ?
(a) [−5, −3]
(c) (5, ∞)
(b) [3, 7]
(d) There are no solutions
log(x − 3) + log x = 1 → log x(x − 3) = 1 → 10log x(x−3) = 101 → x2 − 3x = 10 → x2 − 3x − 10 = 0 →
(x + 2) (x − 5) = 0 → x = −2 (reject) or x = 5.
10. Solve for x: ln(2 log2 x) = 3
√
(a) 2e3
(b) 2e3
(c) e6
ln(2 log2 x) = 3 → eln(2 log2 x) = e3 → 2 log2 x = e3 → log2 x =
e3
2
→ 2log2 x = 2
(d) e8
e3
2
3 12
√
= 2 e3
→ x = 2e
11. Which of the following intervals contains a solution of the equation log2 (2x + 1) − log2 (x − 3) = log2 x ?
(a) (−∞, 0)
(b) (3, 11)
log2 (2x + 1) − log2 (x − 3) = log2 x → log2
x2 − 5x − 1 = 0 → x =
√
5− 29
2
(c) (−1, 3)
2x+1
x−3
(reject) or x =
= log2 x →
√
5+ 29
.
2
2x+1
x−3
(d) There are no solutions
= x → 2x + 1 = x2 − 3x →
12. Match the following functions to their graphs:
A. y = 2x
B. y = 2−x
C. y = −2x
D. y = −2−x
A
B
C
D
E
F
G
H
E. y = 2x − 1
F. y = 2x−1
G. y = 21−x
H. y = 1 − 2x
7
5
1
4
3
2
8
6
13. Match the following functions to their graphs:
A. y = log2 x
B. y = log2 (−x)
C. y = − log2 x
D. y = − log2 (−x)
A
B
C
D
E
F
G
H
E. y = log2 x − 1
F. y = log2 (x − 1)
G. y = log2 (1 − x)
H. y = 1 − log2 x
3
2
4
7
1
6
8
5
Answer questions 14–23 by “True” or “False”. Assume that all variables represent positive quantities.
14. log x = ln x False.
15. log xy = log x + log y True.
16. log(x − y) = log x − log y False.
17.
log2 x
log2 y
= log2 (x − y)
False.
18. log5 xx = x log5 x True.
19. log3 x =
log2 x
log2 3
20. log2 x =
ln x
log 2
True.
False.
21. If f (x) = 5x then f −1 (x) = log5 x True.
√
22. log2 ( 3 2) =
1
3
True.
23. ex = ln x False.
PART II. (Show all your work)
24. Given f (x) =
y=
2x−1
x+2
2x−1
x+2 ,
find f −1 (x) and the range of f .
→ y (x + 2) = 2x − 1 → 2y + xy = 2x − 1 → xy − 2x = −2y − 1 → (y − 2) x = −2y − 1
−1
−1 = Rf = {x : x 6= 2}
→ x = − 2y+1
(x) = − 2x+1
y−2 → f
x−2 . Now, Df
25. What is the domain of G(x) = log
Solve
x2 +7x+10
x−2
> 0:
x2 +7x+10
x−2
x2 +7x+10
x−2
>0→
?
(x+5)(x+2)
x−2
>0
So Domain of G = (−5, −2) ∪ (2, ∞)
26. Solve for x: e2x−1 = 2x+1
e2x−1 = 2x+1 → ln e2x−1 = ln 2x+1 → 2x − 1 = (x + 1) ln 2 → 2x − x ln 2 = 1 + ln 2 →
(2 − ln 2) x = 1 + ln 2 → x =
1+ln 2
2−ln 2
27. Solve for x: log5 (9x + 7) − log5 (x − 1) = 2
log5 (9x + 7) − log5 (x − 1) = 2 → log5
9x+7
x−1
= 2 → 5log5
9x+7
x−1
= 52 →
9x+7
x−1
= 25 →
9x + 7 = 25x − 25 → 16x = 32 → x = 2
28. Find the interest rate at which $50,000 should be invested for 30 years compounded continuously for the account
to grow to $1,000,000.
Solve for r: 50000e30r = 1000000 → e30r = 20 → ln e30r = ln 20 →
30r = ln 20 → r =
ln 20
30
≈ 9. 985 8 %
TEST 4 ANSWERS
1. B and D
15. True
2. C and D
16. False
3. D
17. False
4. C
18. True
5. D
19. True
6. A
20. False
7. B
21. True
8. C
22. True
9. B
23. False
10. A
24. f −1 (x) = − 2x+1
x−2 , Rf = {x : x 6= 2}
11. B
25. Domain of G = (−5, −2) ∪ (2, ∞)
A
B
C
D
E
F
G
H
7
5
1
4
3
2
8
6
26. x =
A
B
C
D
E
F
G
H
27. x = 2
3
2
4
7
1
6
8
5
12.
1+ln 2
2−ln 2
13.
28. r =
14. False
ln 20
30
≈ 9. 985 8 %