math 112 spring 2016 midterm 2 review problems page 1 ...math.arizona.edu/~algebra/math112/s16...

Math 112 – Spring 2016 Midterm 2 Review Problems

1. Solve the inequality.

01032 xx

The solution is:

(A) ,25,

(B) ,52,

(C) 2,5

(D) 5,2

(E) None of these

2. Which one of these equations represents y as a function of x?

(A) 0422 yx (B) 03 2 xy

(C) 25 43 yx (D) 105 2 yx

(E) 723 3 xy


3. Which one of the following graphs DOES NOT represent a function?

(A) (B)

(C) (D)

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6


4. Which one of the tables below represents y as a function of x?

(A)

x 2 0 1 0 4

y 5 1 3 1 5

(B)

x 2 2 0 2 4

y 5 4 3 2 1

(C)

x 1 2 1 4

y 0 3 2 1

(D)

x 1 2 1 4

y 3 3 3 3

(E)

x 1 2 1 4

y 3 2 1 0

5. Given 342 2 rrrf , evaluate 21 tf .

(A) 742 2 tt (B) 12 2 t (C) 742 2 tt

(D) 542 2 tt (E) 32 2 t

6. Which one of the following points lies on the graph of the equation 52 xy ?

(A) 5,0 (B) 3,2 (C) 0,5 (D) 3,2


7. For 492 xxxf , find and simplify h

xfhxf )()( .

(A) h

hxhxh 89182 2 (B)

h

xhh 8182

(C) 92 hx (D) 12 hx

(E) 92 x

8. Determine the domain of 45

122

xx

xxg .

(A) 4

1x

(B) 2

1,1,4 x

(C) 2

1,4,1 x

(D) 4,1x

(E) 1,4 x

9. Determine the zero(s) of 45

122

xx

xxr .

(A) 2

1x only

(B) 4x , 1 , and 2

1 only

(C) 4x and 1 only

(D) 4

1x only

(E) 2

1x and

4

1only


10. Determine the domain of xxh 73 .

(A)

,

7

3 (B)

,

7

3 (C) ,0

(D)

7

3, (E)

7

3,

11. Determine the domain of x

xxf

3

2. (Hint: Determine the domain of the numerator and

denominator separately first.)

(A) ,3

(B) 3,0

(C) 3,0

(D) 3,0

(E) None of these

12. Which one of the following has a domain of all real numbers except 4x ?

(A) 4 xy (B) 16

12

x

y (C) 4 xy

(D) 82

3

x

xy (E)

x

xy

4


13. The graph of xgy is shown below. Use the graph to determine the domain and range of

g.

xg

(A) Domain: ),( Range: ),4[

(B) Domain: ),2.2[ Range: ),4[

(C) Domain: ),4[ Range: ),2.2[

(D) Domain: ),2.2[ Range: ),2.2[

(E) Domain: ),4[ Range: ),(

14. The graph of xfy is shown below. Use the graph to determine the value of 0f .

xf

(A) 2 (B) 2 (C) 0 (D) 3 (E) 5

-4

-2

0

2

4

6

8

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6


15. Identify the open interval(s) where the graph of xTy is decreasing.

(A) 2, and 4,0 only (B) 6,2 only

(C) 4,0 only (D) 2, and 2,0 only

(E) 2, only

16. Identify the open interval(s) where the graph of xTy is positive.

(A) 2, only (B) 0,2 and 6,4 only

(C) 0,2 only (D) 2, and 0,2 only

(E) 2, and 2,2 only

-3

-2

-1

0

1

2

3

4

5

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

-3

-2

-1

0

1

2

3

4

5

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7


17. Given that xLy is an odd function, complete the table of values:

x 3 2 1 1 2 3

xL 6 2 5

The missing values, from left to right, are:

(A) 5

1,

2

1,

6

1 (B) 5 , 2 , 6 (C) 5, 2, 6

(D) 5

1 ,

2

1 ,

6

1 (E) None of these

18. Given that xLy is an even function, complete the table of values:

x 3 2 1 1 2 3

xL 6 2 5

The missing values, from left to right, are:

(A) 5 , 2 , 6 (B) 5

1,

2

1,

6

1 (C) 5, 2, 6

(D) 5

1 ,

2

1 ,

6

1 (E) None of these


19. Determine which one of the following graphs represents an even function.

(A) (B)

(C) (D)

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6


20. Determine which one of the following graphs represents an odd function.

(A) (B)

(C) (D)

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6


21. Determine which one of the following represents an odd function.

(A) 1.03 xxg

(B) 435 xxf

(C) 13 mmmR

(D) ttth 242 3

(E) 23 xxF

22. Evaluate 0g given that:

1for 2

1for 42 2

xx

xxxxg

(A) 1 (B) 0 (C) 2 (D) 0 and 2 (E) None of these

23. Determine the domain and range of

1for 2

1for 1

xx

xxxf .

(A) Domain: , (B) Domain: ,

Range: ,12, Range: ,

(C) Domain: 1,1 (D) Domain: ,11,

Range: ,1 Range: ,12,

24. Determine the range of the function given by

15

112

xx

xxxg

(A) ]4,(

(B) ]3,(

(C) )3,(

(D) )4,(

(E) ),(


25. Which one of the following is the graph of the function

1 if3

1 if2

x

xxxf ?

(A) (B)

(C) (D)

-5

-4

-3

-2

-1

0

1

2

3

4

5

-5 -4 -3 -2 -1 0 1 2 3 4 5

-5

-4

-3

-2

-1

0

1

2

3

4

5

-5 -4 -3 -2 -1 0 1 2 3 4 5

-5

-4

-3

-2

-1

0

1

2

3

4

5

-5 -4 -3 -2 -1 0 1 2 3 4 5

-5

-4

-3

-2

-1

0

1

2

3

4

5

-5 -4 -3 -2 -1 0 1 2 3 4 5


26. Determine the rule that defines the piecewise-defined function graphed below.

(A)

53 if2

1

30 if1

xx

xx

xf (B)

53 if2

1

2

1

30 if1

xx

xx

xf

(C)

53 if2

1

2

1

30 if1

xx

xx

xf (D)

53 if22

1

30 if1

xx

xx

xf

27. If 5,2 is a point on the graph of xry , which one of the following points MUST be

on

the graph of 13

1 xry ?

(A)

4,

3

2 (B) 5,1 (C)

3

5,3 (D)

3

5,1 (E)

5,

3

1

-3

-2

-1

0

1

2

3

4

5

-3 -2 -1 0 1 2 3 4 5


28. Which one of the following transforms the graph of xfy with a vertical compression

and a horizontal stretch?

(A) xfy 23

(B) xfy 23

1

(C) xfy 3

(D)

xfy

2

13

(E)

xfy

2

1

3

1

29. Suppose xgy has a domain of ]10,4( . What is the domain of 32 xgy ?

(A) ]7,7(

(B) ]12,2(

(C) ]13,6(

(D) ]13,1(

(E) ]8,6(

30. Suppose the graph of 2xy is transformed into 257

2 xy . Which one of the

following describes the transformations performed?

(A) Horizontal compression, reflect over the x-axis, shift right

(B) Vertical compression, reflect over the x-axis, shift left

(C) Vertical compression, reflect over the y-axis, shift left

(D) Vertical compression, reflect over the x-axis, shift right

(E) Horizontal expansion, reflect over the y-axis, shift left


31. Consider the table of values for the function xhy given below.

x 4 1 6 10

xh 3 8 4 5

Which one of the following tables of values represents 42 xhy ?

(A)

x 8 2 12 20

42 xh 1 12 0 1

(B)

x 12 2 8 16

42 xh 3 8 4 5

(C)

x 8 2 12 20

42 xh 7 4 8 9

(D)

x 4 1 6 10

42 xh 2 20 4 6

(E)

x 2 0.5 3 5

42 xh 1 12 0 1


32. Consider the graph of xfy given below,

Which one of the following represents the graph of xfy 2 ?

(A) (B)

(C) (D)

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6


33. The graph of 2 xy can be obtained from the graph of xy by

(A) Shifting up 2 units and reflecting over the x-axis.

(B) Shifting right 2 units and reflecting over the x-axis.

(C) Shifting left 2 units and reflecting over the y-axis.

(D) Shifting right 2 units and reflecting over the y-axis.

(E) Shifting left 2 units and reflecting over the x-axis.

Let 2

1

xxf and 12 xxg . Use these functions to answer the next two questions.

34. Evaluate and simplify xg

f

.

(A) 12

2

x

x

(B) 24

1

xx

(C) 2

2 1

x

x

(D) 24 xx

(E) None of these

35. Evaluate and simplify xgf .

(A) 11

4

x

(B) 12

12

x

(C) 2

2 1

x

x

(D) 1

14 x

(E) 12

124 xx

36. Consider the functions 32 xxS and 1

1

xxT . Find 3TS .


(A) 3

(B) 2

3

(C) 1

(D) 2

1

(E) 2

Consider the functions xf and xg represented in the tables below. Use these tables to

answer the next three questions.

x xf x xg

0 2 0 2

1 3 1 2

2 1 2 1

37. Evaluate 2gf .

(A) 0

(B) 1

(C) 2

(D) 3

(E) Undefined

38. Evaluate 0fg .

(A) 0

(B) 1

(C) 2

(D) 3

(E) 4

39. Evaluate 1gf .

(A) 1 (B) 0

(C) 1

(D) 2

(E) 3

40. Determine the domain of xN

M

if

xxM

1 and 62 xxxN .


(A) ),2()2,0()0,3()3,(

(B) ),2()2,3()3,(

(C) ),0()0,(

(D) ),3()3,0()0,2()2,(

(E) ),3()3,2()2,(

41. Determine the domain of xTR if 4

1

xxR and 3 xxT .

(A) ,44, (B) ,1313,3

(C) ,44,3 (D) ,44,3

(E) ,1313,3

42. Determine functions xf and xg so that 22

1

xxgf . Which one of the

following could NOT be choices for xf and xg ?

(A) x

xf1

and 22 xxg

(B) 2

1

xxf and 2 xxg

(C) 2

1

xxf and 2xxg

(D) 21

1

xxf and 1 xxg

43. Suppose 34 xxh , where xgfxh . If xxf 4 , which one of the

following is xg ?

(A) 3xg

(B) 3xg

(C) 3 xxg

(D) 3 xxg

(E) 3 xxg

Use the graphs of the functions xMy and xNy given below to answer the following

questions.


xM xN

44. Determine the domain of xNM .

(A) 5,3 (B) 4,4 (C) 4,3

(D) 5,4 (E) 5,33,4

45. Determine the domain of xN

M

.

(A) 5,3 (B) 4,4 (C) 4,3

(D) 5,4 (E) 4,22,3

46. Determine a formula for the inverse function for 72 3 xxg .

(A) 72

1 31 xxg (B) 31 72

1 xxg

(C) 31

2

7 x

xg (D) 7231 xxg

(E) 72

13

1

xxg

47. Determine a formula for the inverse function for 32

14

x

xxh .

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6


(A) 14

321

x

xxh (B)

42

131

x

xxh

(C) 42

131

x

xxh (D)

32

141

x

xxh

(E) None of these

48. Use the graph of xfy given below to determine the RANGE of the inverse function

xfy 1 .

(A) 6,4 (B) 5,3 (C) 4,6

(D) 3,5 (E) None of these

49. Use the graph of xfy given below to determine the graph of the inverse function

xfy 1 .

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6


(A) (B)

(C) (D)

50. The function xR is described by the set of ordered pairs below.

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6


x 1 2 3 4

xR 6 5 9 8

Which one of the following tables describes xR 1 ?

(A)

x 6 5 9 8

xR 1 1 2 3 4

(B)

x 1 2 3 4

xR 1 6

1

5

1

9

1

8

1

(C)

x 1 2 3 4

xR 1 6 5 9 8

(D)

x 1 2 3 4

xR 1 6 5 9 8

(E)

x 1 2 3 4

xR 1 6 5 9 8

51. If xf is a one-to-one function and 72 f , what point MUST be on the graph of

xfy 1 ?


(A) 7,2 (B)

7,

2

1 (C)

7

1,

2

1 (D) 2,7 (E)

7

1,2

52. The cost (in dollars) of introducing a recycling program to p% of the population in a

particular town can be modeled by:

p

pC

100

000,30 1000 p

Determine the percent of the population the town can put on the recycling program if the

budget for the program is $500,000. Round your answer to the nearest 0.1%

(A) 94.3%

(B) 92.6%

(C) 87.8%

(D) 85.2%

(E) 79.9%

53. The cost (in dollars) of introducing a recycling program to p% of the population in a

particular town can be modeled by:

p

pC

100

000,30 1000 p

Interpret the meaning of 5.62000,501 C .

(A) For a cost of $50,000, one can expect 62.5% of the population to be put on the program.

(B) For a cost of $62,500, one can expect 50,000 people to be put on the program.

(C) For a cost of $62.50, one can expect 50% of the population to be put on the program.

(D) For a cost of $50,000, one can expect 62,500 people to be put on the program.

Consider the functions xf and xg represented in the tables below. Use these tables to

answer the next two questions.


x xf x xg

1 2 4 5

2 3 5 3

3 5 7 1

54. Evaluate 21f .

(A) 5

(B) 4

(C) 3

(D) 2

(E) 1

55. Evaluate 31 fg .

(A) 1

(B) 3

(C) 4

(D) 5

(E) 7

56. Consider the graph of xCy given below.


What is the value of 31C ?

(A) 3.5 (B) 3 (C) 2 (D) 75.2 (E) 4

57. If 86 3 xxP , evaluate 41P .

(A) 376 (B) 8 (C) 3

56 (D) 3 2 (E) None of these

58. Determine xg 1 for 1

4

x

Axxg , where A is a constant.

(A) 4

11

Ax

xxg (B)

Axg

41

(C)

x

xAxg

411 (D)

Ax

xxg

41

(E) None of these

59. Determine whether 21003284 xxf has a maximum or minimum and then determine

the maximum or minimum value.

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6


(A) Max = 100 (B) Min = 100 (C) Min = 8 (D) Max = 52 (E) Max = 84

60. Consider the function 983002 xxf . Which one of the following statements is

TRUE?

(A) The graph opens up. (B) The graph opens down.

(C) The y-coordinate of the vertex is 9. (D) The graph has a minimum at the vertex.

61. Determine the formula for the quadratic function in standard form for the parabola graphed

below.

(A) 122 xy

(B) 1222 xy

(C) 122 xy

(D) 1242 xy

(E) 1242 xy

62. Determine the formula for the quadratic function in standard form that fits the data exactly

-5

-4

-3

-2

-1

1

2

3

4

5

-2 -1 1 2 3 4 5


t 10 20 30 40 50

tH 7 2 7 22 47

(A) 2)20(20

1)( 2 ttH

(B) 20)2(64

13)( 2 ttH

(C) 7)10(20

1)( 2 ttH

(D) 10)7(169

8)( 2 ttH

(E) None of these

63. Determine the x-intercepts of 21121 2 xxxf . The largest x-intercept is:

(A)

0,

3

2 (B)

0,

7

1 (C)

0,

3

2

(D)

0,

7

1 (E) None of these

64. Determine the vertex for the quadratic function 282 xxxg .

(A) 112,6 (B) 32,6 (C) 25,3

(D) 22,3 (E) 50,3

65. Determine a possible formula for the quadratic function that passes through the points 0,4 ,

0,3 , and 2,1 .

(A) 346

1 xxy (B) 34

5

1 xxy

(C) 34 xxy (D) 34 xxy

(E) 342 xxy

66. Determine a possible formula for the quadratic function that has a vertex at 3,5 and

passes through the point 7,1 .


(A) 354

1 2 xy (B) 35

19

7 2 xy (C) 35

9

1 2 xy

(D) 3539

7 2 xy (E) 354

2 xy

67. A mathematical model for the profit, P, in thousands of dollars, that a company makes in one

year from producing x units of its product is:

750,896002 xxxP .

What is the maximum profit the company can make?

(A) $200,000

(B) $250,000

(C) $275,000

(D) $300,000

(E) $325,000

68. A woman is shooting a basketball toward a basketball hoop. The starting height of the ball is

5.5 feet. The path of the basketball is parabolic in shape, and the ball reaches its maximum

height of 11.5 feet when the ball is 10 feet from the player. The ball hits the front of the rim,

which is 10 feet high. Determine the horizontal distance from the shooter to the rim. The

distance is:

(A) Less than 10 feet

(B) Between 10 and 12 feet

(C) Between 12 and 14 feet

(D) Between 14 and 16 feet

(E) More than 16 feet

69. A homeowner wants to fence in a rectangular garden next to a very long stream. Since the

garden will border the stream, that side of the garden will not need fencing. What is the

largest area that can be enclosed with 220 feet of fencing?

(A) 2,200 sq ft

(B) 6,050 sq ft

(C) 8,225 sq ft

(D) 12,100 sq ft

(E) None of these

70. A horticulturist has determined that the number of inches a young redwood tree grows in one

year is a function of the annual rainfall, r (in inches), given by


102.0 2 rrrg

What is the maximum number of inches a young redwood can grow in a year? The

maximum number of inches is:

(A) less than 12

(B) between 12.9 and 13.1

(C) between 13.2 and 13.6

(D) between 14.4 and 15.3

(E) between 24.7 and 25.2

71. When an apple orchard owner plants 65 trees on an acre of ground, he gets an average yield

of 1500 apples per tree per year. For each additional tree planted per acre, the annual yield

per tree drops by 20 apples. Calculate the number of trees that should be planted per acre to

produce the maximum crop of apples and use that value to determine the maximum yield?

The maximum yield is:

(A) 7,000 apples

(B) 30,000 apples

(C) 97,500 apples

(D) 98,000 apples

(E) 100,000 apples

Bob’s high school is performing a musical. If each ticket is priced at $9.50 then 600 tickets will

be sold. For each $0.10 increase in price, 6 fewer tickets will be sold. Use this information to

answer the next two questions.

72. Determine the price that should be charged for each ticket to maximize the revenue. (Hint:

Revenue is quantity times price.)

(A) Less than $9.60

(B) Between $9.60 and $9.80

(C) Between $9.80 and $10.00

(D) Between $10.00 and $10.20

(E) More than $10.20

73. Determine the maximum possible revenue from the musical ticket sales.

(A) More than $6100

(B) Between $6000 and $6100

(C) Between $5900 and $6000

(D) Between $5800 and $5900

(E) Less than $5800


A company has determined that the demand function for a dining room table is given by

xp2

11200 where p is the price per table and x is the number of tables sold. The fixed cost

of producing a line of dining room tables is $100,000 and each table costs $200 to make. Use

this information to answer the next three questions.

74. Determine a simplified formula for the revenue function. (Hint: Revenue is quantity times

price.)

(A) xxR2

11200 (B) 2

2

11200 xxR (C) 2

2

11200 xxxR

(D) xxR2

31200 (E) 2

2

31200 xxxR

75. Determine the quantity that should be manufactured to maximize revenue. The quantity is:

(A) Less than 950 tables

(B) Between 950 and 1050 tables

(C) Between 1050 and 1150 tables

(D) Between 1150 and 1250 tables

(E) More than 1250 tables

76. Determine the quantity that should be manufactured to maximize profit. The quantity is:

(A) Less than 950 tables

(B) Between 950 and 1050 tables

(C) Between 1050 and 1150 tables

(D) Between 1150 and 1250 tables

(E) More than 1250 tables


Question Answer Question Answer

1 B 39 C

2 E 40 A

3 B 41 B

4 D 42 C

5 E 43 D

6 D 44 C

7 C 45 E

8 E 46 C

9 A 47 B

10 A 48 A

11 C 49 C

12 D 50 A

13 C 51 D

14 B 52 A

15 A 53 A

16 E 54 E

17 B 55 C

18 C 56 C

19 A 57 D

20 C 58 D

21 D 59 E

22 C 60 B

23 A 61 E

24 D 62 A

25 C 63 A

26 B 64 E

27 D 65 A

28 E 66 A

29 B 67 B

30 B 68 D

31 E 69 B

32 A 70 C

33 E 71 D

34 B 72 B

35 E 73 E

36 E 74 C

37 D 75 D

38 B 76 B

Spring2016Midterm2StudyGuideAnswers

math 112 spring 2016 midterm 2 review problems page 1 ...math.arizona.edu/~algebra/math112/s16...

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