Math problems. Intro to calculus

ALGEBRA REVIEW

Name

Student Code

Teaching Assistant

Instructor

1

1. The graph of a function f is given below on the left. After transformation of the graph of f we obtain the graph g on the right. From the answer choices below select the one which best represents the graph of g.

f g

(a) g(x) = |f(x2 )| (b) g(x) = −2f(|x|) (c) g(x) = 4|f(x)| (d) g(x) = −|2f(x)| (e) g(x) = −2f(−|x|)

2. Find the solution set for the inequality

−1

6 x ≥ 5

4 +

7

3 x

.

(a) x ∈ [2,+∞)

(b) x ∈ (−∞,−2]

(c) x ∈ ( −∞,−1

2

] (d) x ∈

[ −1

2 ,+∞

) (e) x ∈ [−2,+∞)

3. logπ(e) = x implies that

(a) eπ = x

(b) ex = π

(c) πx = e

(d) πe = x

(e) xπ = e

2

4. Write the inequality −8 ≤ x < 5 using interval notation.

(a) x ∈ (−8, 5]

(b) x ∈ [−8, 5)

(c) x ∈ (−∞, 5)

(d) x ∈ [−8, 5]

(e) none of the other choices

5. Solve for x: a

x = c− b

x .

(a) c(a+ b)

(b) ab

c(a+ b)

(c) a+ b

c

(d) − ab

c(a+ b)

(e) c

a+ b

6. Find the equation of the line perpendicular to the line y = −4x− 5 containing the point (2, 1).

(a) −x+ 4y = 2

(b) 4x− y = 7

(c) −4x+ y = −7

(d) x− 4y = 2

(e) 4y − x = 3

3

Problems 24-25 refer to the graph below.

7. The function f has codomain (range)

(a) y ∈ (2, 0)

(b) y ∈ [−2, 5]

(c) y ∈ (−2, 3)

(d) y ∈ [−2, 3]

(e) y ∈ (−∞,+∞)

8. The function f is negative on the interval(s)

(a) x ∈ (−4, 2)

(b) x ∈ [−4, 0) ∪ (4, 6)

(c) x ∈ (0, 4)

(d) x ∈ (−∞,+∞)

(e) f is never negative

9. Let f(x) = √ x. Apply the following sequence of transformations to the graph of f .

(i) Shift down 3 units.

(ii) Reflect about the x-axis

(iii) Shift left 8 units.

Find the function p(x) that represents the resulting graph.

(a) p(x) = √ x+ 8 + 3

(b) p(x) = − √ x− 8− 3

(c) p(x) = − √ x+ 8− 3

(d) p(x) = √ −x− 8 + 3

(e) p(x) = − √ x+ 8 + 3

4

10. For the graph of g below, define g piecewise.

(a) g(x) =

 3 4x+ 4 if −3 ≤ x ≤ 0

3 2x if 0 < x ≤ 3

(b) g(x) =

 4 3x+ 4 if −3 ≤ x ≤ 0

2 3x+ 2 if 0 < x ≤ 3

(c) g(x) =

 4 3x+ 4 if −3 ≤ x < 0

2 3x if 0 ≤ x ≤ 3

(d) g(x) =

 4 3x− 4 if −3 ≤ x ≤ 0

2 3x if 0 < x ≤ 3

(e) g(x) =

 4 3x+ 4 if −3 ≤ x ≤ 0

2 3x if 0 < x ≤ 3

11. Find the equation of the line containing the points (−6, 3) and (8, 8).

(a) y = − 5 14x+ 36

7

(b) y = 5 14x+ 36

7

(c) y + 3 = 5 14x(x+ 6)

(d) y = 14 5 x−

36 7

(e) y = 14 5 x+ 36

7

12. Find the center (h, k) and radius r of the circle (x− 2)2 + (y + 8)2 = 36.

(a) (−8, 2), r = 36

(b) (−2, 8), r = 6

(c) (−8, 2), r = 6

(d) (2,−8), r = 36

(e) (2,−8), r = 6

5

13. The graph of a function f is given below on the left. After transformation of the graph of f we obtain the graph g on the right. From the answer choices below select the one which best represents the graph of g.

f g

(a) g(x) = f(x− 2)− 1

(b) g(x) = f(x+ 2)− 1

(c) g(x) = f(x− 2) + 1

(d) g(x) = −f(x+ 2)− 1

(e) g(x) = f(x+ 2) + 1

14. Which of the following relations, graphs, or correspondences represents a function?

A. x2 − 5y2 = 1 B. {(−1, 8), (1, 5), (5,−5), (7,−1)}

C. D. xy3 = −6

E. F.

(a) C, B, E (b) B, D,C (c) A, B, C, D, (d) B, D, E (e) A, D, E

6

Questions 32-35 refer to the following function: f(x) = −x2 + 2x+ 5

15. The function f(x) = −x2 + 2x+ 5 opens

(a) upwards

(b) to the left

(c) to the right

(d) downwards

(e) diagonally

16. The vertex of f(x) = −x2 + 2x+ 5 is

(a) (-1,2)

(b) (-2,-3)

(c) (-1,4)

(d) (1,8)

(e) (1,6)

17. The y-intercept of f(x) = −x2 + 2x+ 5 is

(a) (5,0)

(b) (0,-5)

(c) (0,5), (0,-5)

(d) (0,5)

(e) (0,7)

18. The x-intercept(s), in simplified form, of the function f(x) = −x2 + 2x+ 5 is/are

(a) (

1± 2 √

6, 0 )

(b) ( − 1±

√ 6 )

(c) (

1± √

6, 0 )

(d) (−2±

√ 24

2 , 0 )

(e) (1,6)

19. If f(x) is a one to one function with the point (2,−1 3) lying on the graph of f(x), then the inverse function

f−1(x) necessarily has which point lying on it’s graph?

(a) (12 ,−3)

(b) (−3, 12)

(c) (−2, 13)

(d) (−1 3 , 2)

(e) (2,−1 3)

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20. The function h(x) = (x+ 4)(x+ 2)(x− 2)2 is positive in the interval

(a) x ∈ (−∞,−4)

(b) x ∈ (−∞,−4) ∪ (−2, 2) ∪ (2,+∞)

(c) x ∈ (−∞,+∞)

(d) x ∈ (−4,−2)

(e) (-3,2)

21. Find the quadratic function with graph given below.

(a) f(x) = −1 4(x− 2)2 + 3

(b) f(x) = −(x− 2)2 + 3

(c) f(x) = (x+ 2)2 − 3

(d) f(x) = 4(x+ 2)2 − 3

(e) f(x) = 1 4(x− 2)2 − 3

22. For the given functions f(x) = 2x and g(x) = 3x2 + 1, find (g ◦ f)(−2).

(a) 26

(b) -52

(c) 25

(d) 49

(e) 56

Questions 40-44 refer to the graph of the function g(x) = − log5(−x+ 2).

23. The codomain (range) of the function g(x) consists of all

(a) y ∈ (−2,+∞)

(b) y ∈ (−∞,+∞)

(c) y ∈ (−∞,−2)

(d) y ∈ [0,+∞)

(e) y ∈ (−∞,−2]

8

For problem 41, select the choice that best describes the properties of the graph of g(x).

24. The function g(x) has asymptote given by the line .

(a) vertical, x = −2

(b) horizontal, y = 0

(c) vertical x = 0

(d) horizontal, y = 2

(e) vertical, x = 2

25. The function g(x) has x-intercept

(a) (1, 0)

(b) (2, 0)

(c) (5, 0)

(d) (0, 0)

(e) g has no x-intercept

26. The function g(x) is increasing on the interval

(a) x ∈ (2,+∞)

(b) x ∈ (−∞,+∞)

(c) x ∈ (−∞, 2)

(d) x ∈ [2,+∞)

(e) g is never increasing

27. The function g(x) has inverse function

(a) g−1(x) = 5−x − 2

(b) g−1(x) = 5−x−2

(c) g−1(x) = −5−x + 2

(d) g−1(x) = 1

− log5(−x+ 2)

(e) g−1(x) = − log5(2− x)

28. The circle above has center

(a) (−3, 2) (b) (3, 2) (c) (32 , 2) (d) (52 , 2) (e) (2, 52)

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29. ln(y) = 2 ln(x)− ln(x+ 1)− ln(C) implies that

(a) y = −Cx2(x+ 1)

(b) y = C(x+ 1)

x2

(c) y = x2

C(x+ 1)

(d) y = ln(Cx)

(e) x+ 1

Cx2

30. If 5x = 2 what is 53x?

(a)2

(b)4

(c)8

(d)16

(e)32

31. Solve log4 (x− 1) = 2 for x

(a) x = 17

(b) x = 8

(c) x = 2, x = −9

(d) x = 9

(e) x = −2

32. The x-intercept of f(x) = 30− 12ex is

(a) (e √ 2, 0)

(b) (ln( 5

2 ), 0)

(c) (40, 0)

(d) (60, 0)

(e) ( ln(30)

ln(12) , 0)

10

Problems 50-51 refer to the graph below.

33. The function f is strictly increasing on the interval(s)

(a) x ∈ −2, 4)

(b) x ∈ (3, 6)

(c) x ∈ (−6,−4)

(d) x ∈ (−4, 0)

(e) f is never increasing

34. The function f is constant on the interval(s)

(a) x ∈ (0, 3)

(b) x ∈ (−4, 0)

(c) x ∈ (−6,−4)

(d) x ∈ [3, 6]

(e) f is never constant

35. Solve the system of equations

 −x+ 2y = 7

3x− 5y = 2

(a) (39, 23)

(b) (2,−2)

(c) (4, 1)

(d) (−39, 23)

(e) (24, 16)

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36. Which graph best represents the graph of f(x) = −3x + 1?

graph 1 graph 2

graph 3 graph 4

graph 5 graph 6

(a) graph 3

(b) graph 2

(c) graph 1

(d) graph 6

(e) graph 5

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37. Which of the following graphs best represents the solution the system of equations

 xy = −4

x2 + y2 = 8

A. B.

C. D.

(a) A

(b) B

(c) C

(d) D

(e) none of the other choices

13

38. Which graph best represents the graph of f(x) = log3(−x+ 1)?

graph 1 graph 2

graph 3 graph 4

graph 5 graph 6

(a) graph 3

(b) graph 2

(c) graph 1

(d) graph 6

(e) graph 5

39. Find the quadratic function with graph given below.

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(a) f(x) = −(x− 2)2 + 3

(b) f(x) = −1 4(x− 2)2 + 3

(c) f(x) = 1 4(x− 2)2 − 3

(d) f(x) = (x+ 2)2 − 3

(e) f(x) = 4(x+ 2)2 − 3

Questions 2-5 refer to the function h(x) = −x2(x2 − 81)(x+ 9)

40. The function h(x) is negative in the interval

(a) x ∈ (−∞,−9)

(b) x ∈ (9,+∞)

(c) x ∈ (−∞,+∞)

(d) x ∈ (−∞,−9) ∪ (−9, 0) ∪ (0, 9)

(e) h is never negative

41. The function h(x) is positive in the interval

(a) x ∈ (−∞,−9)

(b) x ∈ (9,+∞)

(c) x ∈ (−∞,+∞)

(d) x ∈ (−∞,−9) ∪ (−9, 0) ∪ (0, 9)

(e) h is never positive

42. Which of the following best describes the end behaviour of h(x)

(a) as x→ −∞, h(x)→ +∞ and as x→ +∞, h(x)→ −∞ (b) as x→ −∞, h(x)→ −∞ and as x→ +∞, h(x)→ +∞ (c) as x→ −∞, h(x)→ −∞ and as x→ +∞, h(x)→ −∞ (d) as x→ −∞, h(x)→ +∞ and as x→ +∞, h(x)→ +∞ (e) none of the other choices

43. Which graph best represents the graph of h(x)?

15

graph 1 graph 2

graph 3 graph 4

(a) graph 1

(b) graph 2

(c) graph 3

(d) graph 4

(e) none of the other choices

16

44. Let f(x) = 3

x+ 2 . Then for x 6= 1,

f(x)− f(1)

x− 1 =

(a) 3

x(x+ 2)

(b) 1

x+ 2

(c) 3

(x− 1)(x+ 2)

(d) −1

x+ 2

(e) −x+ 5

x− 1

45. Let f(x) = 7x+ 4. Then for h 6= 0, f(x+ h)− f(x)

h =

(a) 7 + 14(x+ 4)

h

(b) 0

(c) 7

(d) 7 + 8

h

(e) h+ 8

h

46. Let f(x) = √

2x and a = 2. Geometrically, the difference quotient f(x)− f(a)

x− a is equal to

(a) the slope of the secant line passing through the point (x, √

2x)) and the point (2, 2)

(b) the slope of the tangent line passing through the point (x, f(x)) and the point (2, √

2)

(c) the slope of the secant line passing through the point (−x, f(x)) and the point (2, 2)

(d) the average height of the function between the point (x, f(x)) and the point (2, √

2)

(e) the slope of the secant line passing through the point (−x, √

2x)) and the point (2, 2)

17

Questions 9-17 refer to the function f(x) =

 −x+ 1 if −1 ≤ x < 1

2 if x = 1 x2 if x > 1

47. The domain of f(x) consists of all

(a) x ∈ (−∞,+∞)

(b) x ∈ (−1, 1)

(c) x ∈ [−1, 1]

(d) x ∈ [−1,∞)

(e) none of the above

48. The range of f(x) consists of all

(a) y ∈ (−∞,+∞)

(b) y ∈ [0,∞)

(c) y ∈ (0,∞]

(d) y ∈ (−1, 1)

(e) none of the above

49. The function f(x) is increasing on the interval

(a) x ∈ (−∞,∞)

(b) x ∈ (1,∞)

(c) x ∈ (−∞, 1)

(d) x ∈ (−1, 1)

(e) none of the above

50. The function f(x) is decreasing on the interval

(a) x ∈ (−∞,∞)

(b) x ∈ (1,∞)

(c) x ∈ (−∞, 1)

(e) x ∈ (−1, 1)

(e) none of the above

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51. As x→ +∞, f(x)→ ?

(a) f(x)→ +∞

(b) f(x)→ −∞

(c) f(x)→ 2

(d) f(x)→ −2

(e) f(x)→ 0

52. The function f(x) has x-intercept(s)

(a) (1, 0)

(b) (−1, 0), (1, 0)

(c) (0, 0)

(d) (−1, 0)

(e) none

53. The function f(x) has y-intercept(s)

(a) (0, 1)

(b) (0,−1), (0, 1)

(c) (0, 0)

(d) (0,−1)

(e) none

54. The function f(x) is negative in the interval

(a) x ∈ (−∞,+∞)

(b) x ∈ (−1, 1)

(c) x ∈ (1,∞)

(d) x ∈ (−∞, 1)

(e) never

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55. The function f(x) is positive in the interval

(a) x ∈ (−∞,+∞)

(b) x ∈ (−1, 1)

(c) x ∈ (1,∞)

(d) x ∈ [−1,∞)

(e) never

56. f(34)

(a) 1 4 , 2, 9

16

(b) 2

(c) 9 4

(d) 1 4

(e) 9 16

57. A projectile is thrown upward so that its distance above the ground after t seconds is h(t) = −10t2 + 280t. After how many seconds does it reach its maximum height?

(a) 7 sec

(b) 14 sec

(c) 21 sec

(d) 28 sec

(e) 35 sec

58. You have 120 feet of fencing to enclose a rectangular plot that borders a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area.

(a) length: 60 ft, width: 30 ft

(b) length: 90 ft, width: 30 ft

(c) length: 60 ft, width: 60 ft

(d) length: 30 ft, width: 30 ft

(e) none of the above

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59. The price p (in dollars) and the quantity x sold of a certain product obey the demand equation x = −5p+100, 0 ≤ p ≤ 20. Express the revenue R as a function of x. and find the value of x that maximizes revenue (xmax).

(a) R(x) = p(−5p+ 100), xmax = 25 4

(b) R(x) = x(−5x+ 100), xmax = 100 5

(c) R(x) = x

( x− 100

5

) , xmax = 50

2

(d) R(x) = x

( x− 100

−5

) , xmax = 100

2

(e) none of the above

60. Find the vertex and axis of symmetry for f(x) = −x2 + 12x+ 4

(a) (12, 4); x = 12

(b) (−6,−32); x = −6

(c) (−6,−104); x = −6

(d) (6, 40); x = 6

(e) (6, 40); x = 40

61. Determine the domain and range of f(x) = x2 − 4x+ 4

(a) Domain: (−∞,+∞), Range: [4,+∞)

(b) Domain: (−∞,+∞), Range: [0,+∞)

(c) Domain: (−2,+∞), Range: [0,+∞)

(d) Domain: (2,+∞), Range: [0,+∞)

(e) Domain: (−∞,+∞), Range: [16,+∞)

62. For the function: f(x) =

{ x− 1 if −3 < x < 0 3x− 1 if x ≥ 0

Find f(−3).

(a) − 4

(b) Undefined

(c) − 10

(d) − 2

(e) none of the above

63. For the polynomial f(x) = 2(x+ 5)(x− 6)4 list each real zero and its multiplicity.

(a) − 5,multiplicity 1; 6 multiplicity 4

(b) 5 multiplicity 1; −6 multiplicity 4

(c) − 5 multiplicity 1; 6, multiplicity 4

(d) 5 multiplicity 1; −6, multiplicity 4

(e) 2,multiplicity 1, −5 multiplicity 1; 6, multiplicity 4

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64. Which of the following polynomial functions could have the graph given below?

(A) y = −(x− 4)(x− 2)(x+ 2)

(B) y = (x2 + 1 2)(x2 − 4)(4− x)

(C) y = −1

2 (x2 − 4)2(x− 4)

(D) y = (x2 − 4)(1− x

4 )

(E) y = −1

2 (x2 + 4)(x− 4)

(F) y = 1

2 (x2 − 4)2(x− 4)

(a) F only

(b) B, D, and A

(c) C,E, and F

(d) B and D only

(e) A only

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