Final Exam

Math 1103, Fall 2012

1. Find the slope and y-intercept of the line that is parallel to 2𝑥 + 3𝑦 = 5 and passes
through the point (1, −1)

! !
a. 𝑆𝑙𝑜𝑝𝑒 = ! ; 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 =   !

! !
b. 𝑆𝑙𝑜𝑝𝑒 = − ! ; 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 =   !

! !
c. 𝑆𝑙𝑜𝑝𝑒 = − ! ; 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 =   − !

! !
d. 𝑆𝑙𝑜𝑝𝑒 = − ! ; 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 =   !

e. None of the above

!
2. Find the domain of the function 𝑓 𝑥 = ! ! !!!!
a. (−∞, ∞)
b. 𝑥≠1
c. 𝑥 ≠ −2
d. 𝑥 ≠ 2, −1
e. 𝑥 ≠ −2,1

3. If the point −2,1 is on the graph of 𝑓(𝑥) and 𝑓(𝑥) is known to be odd, what other point
must be on the graph of 𝑓(𝑥)

a. −2, −1
b. 2, −1
c. −2,1
d. 1, −1
e. 0, −1

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4. Find the value of 𝑓 2 − 𝑓(0), if
2 − 𝑥,                                        𝑥 < 1
𝑓 𝑥 =
 𝑥 ! − 𝑥 + 1,                    𝑥 ≥ 1
a. 3
b. −1  
c. 2
d. 0
e. 1

! !
5. If 𝑓 𝑥 = ! + 1 and 𝑔 𝑥 = ! − 1, find 𝑓𝑔 (𝑥).

!
a. !!
−1

b. 1

c. 1 − 𝑥  
 
!
d. !!!
 
 
e. 0

6. The length of a rectangle is 5 units longer than twice its width. Assuming that the width
of the rectangle is w and the area is 𝐴, find the area  as a function of the width.
a. 𝐴 𝑤 = 𝑤 ! + 5𝑤
b. 𝐴 𝑤 = 2𝑤 ! + 5
c. 𝐴 𝑤 = 2𝑤 ! − 5𝑤
d. 𝐴 𝑤 = 2𝑤 ! + 5𝑤
e. None of the above

2
7. 1000 dollars grows to 1 million dollars after 60 years in a bank. If interest is compounded
continuously, what is the rate of interest per year?

a. 1.83%
b. 11.51%
c. 28.13%
d. 3.84%
e. 0.12%

8. Find the sum of all the zeros of the polynomial 𝑓 𝑥 = 𝑥 ! + 2𝑥 ! − 5𝑥 − 6

a. −5

b. −2  

c. 0

d. 2

e. 6

9. The graph of 𝑦 = (𝑥 − 4)! + 5 can be obtained by the transformation of 𝑔 𝑥 = 𝑥 ! .

Which of the following transformations must be used?
I. Move 5 units down.
II. Move 5 units up.
III. Move 4 units down.
IV. Move 4 units left
V. Move 4 units right.

a. V, then II
b. IV, then II
c. III, then I
d. II, then III
e. III, then II

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10. Which of the following functions represents the inverse of the function 𝑓 𝑥 = 3𝑒 !  .
a. 𝑓 𝑥 = 3𝑒 !!
!
b. 𝑓 𝑥 = !! !
!
c. 𝑓 𝑥 = ln  (!)
!
d. 𝑓 𝑥 = ! ln  (𝑥)
!
e. 𝑓 𝑥 = log  (!)

11. The vertex of the parabola 𝑓 𝑥 = 2𝑥 ! − 4𝑥 + 7 is

a. −1  , 13
b. 1  , 7
c. 2  , 5
d. −1  , 5
e. 1  , 5

 
12. Find the horizontal asymptote (HA) and vertical asymptote (VA) of
𝑥! − 4
𝑓 𝑥 =
𝑥(𝑥 + 2)

a. HA: 𝑦 = 1 VA: 𝑥 = 0, 𝑥 = −2

b. HA: 𝑦 = 0 VA: 𝑥 = 0, 𝑥 = −2

c. HA: 𝑦 = 1 VA: 𝑥 = 0

d. HA: 𝑦 = 0 VA: 𝑥 = 0

e. HA: None VA: 𝑥 = 0, 𝑥 = −2

4
13. Find the oblique asymptote of  
𝑥! + 1
𝑓 𝑥 =
𝑥−1

a. 𝑦=1
b. 𝑦 =𝑥−1
c. 𝑦 = 𝑥! + 1
d. 𝑦 =𝑥+1
e. 𝑦=0

! !
14. If 𝑓 𝑥 = ! and 𝑔 𝑥 = 1 − !   , find (𝑔 ∘ 𝑓)(𝑥)

a. 1
b. 1 − 𝑥
!
c. −1 + ! !

! !
d. !
− !!

e. 0

15. What are all the possible rational roots of 𝑓 𝑥 = 6𝑥 ! − 𝑥 ! − 4𝑥 ! − 𝑥 − 2?

! !
a. ±1, ±2, ±3, ±6, ± ! , ± !
! ! ! !
b. ±1, ±2, ± ! , ± ! , ± ! , ± !
! !
c. − ! , !
!
d. −1, !
e. None of the above

5
16. Which of the following intervals represents the solution set to the inequality
!!!
!!!
>3

a. −4, ∞
!"
b. −∞, − !
!"
c. − ! , −4
d. (−4,1)
e. −∞, 3)
 

17. Which of the following statements are true?

I. (ln 𝑥)! = 2 ln 𝑥
II. log ! 3𝑥 ! = 4 log ! (3𝑥)
!"# !
III. log 𝑥 − 𝑦 = !"# !
!
IV. log ! ! = 2 − log ! 4
V. ln 𝑥 ! = 2 ln 𝑥

a. I and II only
b. I, II, and III only
c. I and III only
d. IV and V only
e. I and IV only

18. Solve 𝑙𝑜𝑔(𝑥 − 1) + 𝑙𝑜𝑔(𝑥 + 1) = 0

a. 𝑥 = 2
b. 𝑥 = −1, 𝑥 = 1
c. 𝑥 =1
d. 𝑥 = − 2, 𝑥 = 2
e. 𝑥 =2

6
19. Solve the equation 2!!! = 16!
a. 𝑥 = 0
b. 𝑥 = 1
c. 𝑥 = 2
d. 𝑥 = 3
!
e. 𝑥 = !

!
20. Convert the equation 3!! = !   to logarithmic form

!
a. log ! ( !) = −2
!
b. log ! ( −2) = !
!
c. log !! ( !) = 3
d. log ! (3) = −2
!
e. log ! (−2) = 3
!

!
21. Find the range of the function 𝑓 𝑥 = 5 sin[2(𝑥 + ! )] − 4

a.  [−1,1]
! !
b. [− ! , ! ]

c. [−1,9]
d. [−9,1]
e. None of the above.

7
 !
22. Suppose that sin 𝜃 = ! and 𝜃 is in Quadrant 2. Evaluate cos 𝜃
!!
a. !"

!
b. !"

!"
c. − !

!"
d. !

!"
e. !

23. Find the quadrant in which the terminal side of 𝜃 = 4  𝑟𝑎𝑑𝑖𝑎𝑛𝑠    is located

a. One
b. Two
c. Three
d. Four
e. None of the above.

! !!! !!(!)
24. Find !
if 𝑓 𝑥 = 𝑥 ! + 2𝑥 − 1
a. ℎ + 6
b. 2
c. 2 + ℎ
d. ℎ! + 2ℎ − 1
e. 1

8
25. Find the exact value of 𝑐𝑜𝑡 !! (−1).
!
a. − !
!
b. !
!!
c. !
!!
d. − !

e. None of the above.

! ! !
26. Find the inverse of the function 𝑓 𝑥 = sin(!), where − ! 𝜋 ≤ 𝑥 ≤ ! 𝜋,

!
a. 𝑓 !! 𝑥 = !"#(!!)

b. 𝑓 !! 𝑥 = csc(5𝑥)
!
c. 𝑓 !! 𝑥 = ! sin!! (𝑥)

d. 𝑓 !! 𝑥 = 5sin!! (𝑥)
!
e. 𝑓 !! 𝑥 = sin!! (!!)

!
27. By using sum or difference formulas, cos( ! − 𝑥) can be written as

a. − cos 𝑥

b. sin 𝑥

c. cos 𝑥
d. − sin 𝑥
e. None of the above

9
28. Which of the following is an expression for cos(2𝛼)
a. 1 + 2cos  ! (𝛼)
b. −1 + 2cos ! 𝛼
c. 1 − cos  ! (𝛼)
d. −1 − cos  ! (𝛼)
e. 2cos(𝛼)

29. A 41 meter guy wire is attached to the top of a 34.6 meter antenna and to a point on the
ground. What angle, in degrees, does the guy wire make with the ground?

a. 1°

b. 57.55°
c. 37.65°
d. 45°
e. None of the above.

30. Find an angle 𝜃  between 0∘ and 360∘ that is coterminal with −790∘
a. 𝜃 = 90°
b. 𝜃 = 70°
c. 𝜃 = −70°
d. 𝜃 = 290°
e. None of the above

10
31. The general solution of the equation cos  (2𝜃) = 1 is

a. 𝑘𝜋, where 𝑘 is an integer

b. 0
c. 2𝑘𝜋, where 𝑘 is an integer
!
d. ! + 2𝑘𝜋 , where 𝑘 is an integer
!!
e. !
+ 2𝑘𝜋, where 𝑘 is an integer

32. Simplify
sec 𝑥 − sec 𝑥   . sin! 𝑥    

a. 1
b. sec 𝑥  
c. sin! 𝑥
d. cos ! 𝑥
e. cos 𝑥

33. Find the length of the side 𝑐 in the triangle 𝐴𝐵𝐶 where 𝑎 = 3, 𝑏 = 3 and ∠𝐴𝐶𝐵 = 120°

(Note the figure is not drawn to scale)

a. 27
b. 27

c. 18 − 9 3

d. 18 + 9 3
e. 10

11
34. For what values of 𝑥 in the interval [−2𝜋, 2𝜋] does the graph of 𝑦 = cot(2𝑥) have a
vertical asymptote? (Angles are measured in radians)

a. −2, −1, 0, 1, 2
!! ! ! !!
b. −2𝜋, − !
, −𝜋, − ! , 0, ! , 𝜋, !
, 2𝜋

c. −2𝜋, −𝜋, 0, 𝜋, 2𝜋
!! !!
d. − !
, !

e. −2𝜋, 0, 2𝜋

35. In the figure below, ∠𝐶 = 125°  , 𝐴𝐵 = 8.6  𝑖𝑛𝑐ℎ𝑒𝑠,  and 𝐴𝐶 = 5.7  𝑖𝑛𝑐ℎ𝑒𝑠. Find ∠𝐵 in
degrees.

(Note the figure is not drawn to scale)

a. ∠𝐵 = 29.8°
b. ∠𝐵 = 32.9°
c. ∠𝐵 = 35.7°
d. ∠𝐵 = 38.2°
e. ∠𝐵 = 30.6°

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SOLUTION KEY
1. c
2. e
3. b
4. e
5. a
6. d
7. b
8. b
9. a
10. c
11. e
12. c
13. d
14. b
15. b
16. c
17. d
18. a
19. e
20. a
21. d
22. c
23. c
24. a
25. c
26. d
27. b
28. b
29. b
30. d
31. a
32. e
33. a
34. b
35. b

13