MAC 1105 - Chapters 1 through 6 - Final Exam Review

Name___________________________________

REVIEW DIRECTIONS: Solve each problem.

EXAM DIRECTIONS: The actual exam will be multiple choice. On the exam, you should choose the one alternative that

best completes the statement or answers the question. If a correct answer is not given, choose E.

CALCULATOR NOTE: On the actual exam, you may use only a single-line scientific calculator as specified in the course

syllabus. You may not use a graphing calculator or a calculator on your cell phone on this exam. If you use an

unapproved calculator, it will be considered cheating, your test will be taken away, and you will recieve a score of zero.

Use the graph to determine the x- and y-intercepts.

1)

x-10 -5 5 10

y

10

5

-5

-10

x-10 -5 5 10

y

10

5

-5

-10

1)

2) Solve the linear equation. Then, find the sum of the digits of the answer.

(-7x - 2) - 1 = -6(x - 9)

2)

Solve the equation.

3)2x

5 =

x

3 + 2 3)

First, write the value(s) that make the denominator(s) zero. Then solve the equation.

4)10

5x - 5 +

1

5 =

2

x - 14)

Solve the formula for the specified variable.

5) A = 1

2h(a + b) for b 5)

6) Solve the problem.

You inherit $10,000 with the stipulation that for the first year the money must be invested

in two stocks paying 6% and 11% annual interest, respectively. How much should be

invested in the stock that pays 11% annual interest if the total interest earned for the year is

to be $900?

6)

1

7) Solve the problem.

During a road trip, Tony drove one-third the distance that Lana drove. Mark drives 18

more miles than Lana drove. The total distance they drove on the trip was 298 miles. How

many miles did Mark drive?

7)

Add or subtract as indicated and write the result in standard form.

8) (8 + 4i) - (-3 + i) 8)

Divide and express the result in standard form.

9)3i

7 + 3i9)

Perform the indicated operations and write the result in standard form.

10) (-7 + -64)2 10)

11) Solve the equation by factoring. Then, find the product of the two possible values of x.

6x2 + 23x + 20 = 0

11)

Solve the equation using the quadratic formula.

12) 6x2 = -8x - 1 12)

Solve the radical equation, and check all proposed solutions.

13) 18x - 45 = x + 2 13)

Solve and check the equation.

14) 2x5/2 - 8 = 0 14)

Solve the absolute value inequality. Other than ∅, use interval notation to express the solution set and graph

the solution set on a number line.

15) 8 + 1 - x

2 ≥ 10

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

15)

2

Use the graph to determine the function's domain and range.

16)

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

y6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

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

y6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

16)

Evaluate the piecewise function at the given value of the independent variable.

17) f(x) = x + 5 if x > -4

-(x + 5) if x ≤ -4; f(-6) 17)

Find and simplify the difference quotient f(x + h) - f(x)

h, h≠ 0 for the given function.

18) f(x) = 3x2 18)

Determine the slope and the y-intercept of the graph of the equation.

19) 4x + y - 10 = 0 19)

Use the given conditions to write an equation for the line in the indicated form.

20) Passing through (4, 4) and perpendicular to the line whose equation is y = 2x + 7;

point-slope form

20)

Begin by graphing the standard quadratic function f(x) = x2 . Then use transformations of this graph to graph the given

function.

21) g(x) = - 1

2(x + 5)2 + 1

x-10 -8 -6 -4 -2 2 4 6 8 10

y10

8

6

4

2

-2

-4

-6

-8

-10

x-10 -8 -6 -4 -2 2 4 6 8 10

y10

8

6

4

2

-2

-4

-6

-8

-10

21)

3

Given functions f and g, perform the indicated operations.

22) f(x) = 5x - 3, g(x) = 8x - 8

Find f - g.

22)

For the given functions f and g , find the indicated composition.

23) f(x) = 7x + 15, g(x) = 2x - 1

(f∘g)(x)

23)

Find the domain of the composite function f∘g.

24) f(x) = x + 2, g(x) = 9

x + 1024)

Find the inverse of the one-to-one function.

25) f(x) = 2

3x + 525)

Does the graph represent a function that has an inverse function?

26)

x

y

x

y

26)

Find the distance between the pair of points.

27) (2 10, 3) and (4 10, 6) 27)

Find the midpoint of the line segment whose end points are given.

28) (4

3, -

9

4) and (-

7

3, -

3

4) 28)

Write the standard form of the equation of the circle with the given center and radius.

29) (3, -1); 3 29)

Complete the square and write the equation in standard form. Then give the center and radius of the circle.

30) x2 + y2 + 8x + 12y = 29 30)

Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of

the minimum or maximum point.

31) f(x) = -x2 - 3x - 8 31)

4

Use the Leading Coefficient Test to determine the end behavior of the polynomial function.

32) f(x) = -2x3 - 2x2 - 5x - 4 32)

Find the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses the

x-axis or touches the x-axis and turns around, at each zero.

33) f(x) = 4(x + 2)(x - 4)4 33)

Divide using long division.

34)-8x3 - 22x2 - 31x - 19

4x + 534)

Divide using synthetic division.

35) (x4 + 1296) ÷ (x - 6) 35)

Use the Rational Zero Theorem to list all possible rational zeros for the given function.

36) f(x) = x5 - 4x2 + 5x + 15 36)

Find a rational zero of the polynomial function and use it to find all the zeros of the function.

37) f(x) = x3 - 5x2 + 3x + 1 37)

Find an nth degree polynomial function with real coefficients satisfying the given conditions.

38) n = 3; 3 and i are zeros; f(2) = 25 38)

39) Solve the polynomial equation. In order to obtain the first root, use synthetic division to

test the possible rational roots. Then, find the sum of all the possible values of x.

2x3 - 11x2 + 17x - 6 = 0

39)

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for the given function.

40) f(x) = x7 + x4 + x2 + x + 3 40)

Find the domain of the rational function.

41) g(x) = 2x

(x + 5)(x + 8)41)

5

Use the graph of the rational function shown to complete the statement.

42)

x-10 -8 -6 -4 -2 2 4 6 8 10

y10

8

6

4

2

-2

-4

-6

-8

-10

x-10 -8 -6 -4 -2 2 4 6 8 10

y10

8

6

4

2

-2

-4

-6

-8

-10

As x→3-, f(x)→ ?

42)

Find the vertical asymptotes, if any, of the graph of the rational function.

43) h(x) = x + 2

x2 - 443)

Find the horizontal asymptote, if any, of the graph of the rational function.

44) g(x) = 9x2

3x2 + 144)

Find the slant asymptote, if any, of the graph of the rational function.

45) f(x) = x2 + 3x - 3

x - 945)

Solve the polynomial inequality and graph the solution set on a number line. Express the solution set in interval

notation.

46) x2 + 9x + 18 > 0

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

46)

Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval

notation.

47)-x - 1

x + 9 ≤ 0

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

47)

Solve the problem.

48) The amount of water used to take a shower is directly proportional to the amount of time

that the shower is in use. A shower lasting 24 minutes requires 9.6 gallons of water. Find

the amount of water used in a shower lasting 5 minutes.

48)

6

Solve.

49) The amount of time it takes a swimmer to swim a race is inversely proportional to the

average speed of the swimmer. A swimmer finishes a race in 400 seconds with an average

speed of 3 feet per second. Find the average speed of the swimmer if it takes 300 seconds to

finish the race.

49)

Solve the problem.

50) The pressure of a gas varies jointly as the amount of the gas (measured in moles) and the

temperature and inversely as the volume of the gas. If the pressure is 930 kPa

(kiloPascals) when the number of moles is 4, the temperature is 310° Kelvin, and the

volume is 480 cc, find the pressure when the number of moles is 2, the temperature is

340° K, and the volume is 240 cc.

50)

Use the compound interest formulas A = P 1 + r

n

nt and A = Pert to solve.

51) Find the accumulated value of an investment of $2000 at 10% compounded quarterly for 5

years.

51)

52) Find the accumulated value of an investment of $7000 at 7% compounded continuously for

3 years.

52)

Write the equation in its equivalent exponential form.

53) logb

49 = 2 53)

Write the equation in its equivalent logarithmic form.

54) b4 = 20,736 54)

Find the domain of the logarithmic function.

55) f(x) = ln (9 - x) 55)

Evaluate or simplify the expression without using a calculator.

56) ln e 56)

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate

logarithmic expressions without using a calculator.

57) log3

(27x) 57)

58) loga x4

3x + 5

(x -2)258)

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose

coefficient is 1. Where possible, evaluate logarithmic expressions.

59)1

5[4ln (x + 2) - ln x - ln (x2 - 1)] 59)

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places

60) log0.2

14 60)

7

Solve the equation by expressing each side as a power of the same base and then equating exponents.

61) 2(7 - 3x) = 1

461)

Solve the exponential equation. Express the solution set in terms of natural logarithms.

62) 4 x + 8 = 7 62)

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic

expressions. Give the exact answer.

63) log2

(x + 1) + log2

(x - 5) = 4 63)

64) log8

(5x - 3) = log8

(3x + 1) 64)

Solve.

65) The value of a particular investment follows a pattern of exponential growth. In the year

2000, you invested money in a money market account. The value of your investment t

years after 2000 is given by the exponential growth model A = 2500e0.06t. When will the

account be worth $3805?

65)

Determine whether the given ordered pair is a solution of the system.

66) (3, -6)

2x + y = 12

4x + 2y = 24

66)

Solve the system of equations by the substitution method. Give only the value of x as your answer.

67) x - 5y = -11

-4x - 6y = 18

67)

68) Solve the system by the addition method. Give only the value of x as your answer.

5x + 12y = 12

4x - 6y = -6

68)

Solve the problem.

69) Steve invests in a circus production. The cost includes an overhead of $42,000, plus

production costs of $3000 per performance. A sold-out performance brings in $10,000.

Determine the dollar amount coming in and going out at the break-even point.

69)

Determine if the given ordered triple is a solution of the system.

70) (-4, 2, -3)

x - y + z = -9

x + y + z = -5

x + y - z = 1

70)

71) Solve the system of equations. Give only the values of x and z in your answer.

x + y + z = -3

x - y + 2z = -18

4x + y + z = -12

71)

8

Solve the problem.

72) A basketball player scored 17 points in a game. The number of three-point field goals the

player made was 11 less than three times the number of free throws (each worth 1 point).

Twice the number of two-point field goals the player made was 9 more than the number

of three-point field goals made. Find the number of free-throws and two-point field goals

that the player made in the game.

72)

73) Write the partial fraction decomposition of the rational expression.

x - 8

(x - 2)(x - 4)

73)

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the

constants.

74)2x - 2

(x + 1)(x + 3)274)

75) Write the form of the partial fraction decomposition of the rational expression. It is not

necessary to solve for the constants.

3x - 3

(x - 3)(x2 + x + 7)2

75)

Solve the system by the substitution method.

76) 15x - y = 51

y = x2 + 5

76)

77) xy = 20

x + y = 9

77)

Solve the system by the addition method.

78) x2 + y2 = 64

x2 - y2 = 64

78)

79) x2 - y2 = 25

4x2 + 25y2 = 100

79)

9

Graph the solution set of the system of inequalities or indicate that the system has no solution.

80) -x + 3y < -3

x ≥ -4

x-10 -5 5 10

y

10

5

-5

-10

x-10 -5 5 10

y

10

5

-5

-10

80)

81) x2 + y2 ≤ 100

-10x + 7y ≤ -70

x-10 -5 5 10

y

10

5

-5

-10

x-10 -5 5 10

y

10

5

-5

-10

81)

Solve the problem.

82) A steel company produces two types of machine dies, part A and part B. The company

makes a $2.00 profit on each part A that it produces and a $5.00 profit on each part B that it

produces. Let x = the number of part A produced in a week and y = the number of part B

produced in a week. Write the objective function that describes the total weekly profit.

82)

83) An office manager is buying used filing cabinets. Small file cabinets cost $4 each and large

file cabinets cost $9 each, and the manager cannot spend more than $59 on file cabinets. A

small cabinet takes up 7 square feet of floor space and a large cabinet takes up 10 square

feet, and the office has no more than 86 square feet of floor space available for file cabinets.

The manager must buy at least 7 file cabinets in order to get free delivery. Let x = the

number of small file cabinets bought and y = the number of large file cabinets bought.

Write a system of inequalities that describes these constraints.

83)

10

An objective function and a system of linear inequalities representing constraints are given. Graph the system of

inequalities representing the constraints. Find the value of the objective function at each corner of the graphed region.

Use these values to determine the maximum value of the objective function and the values of x and y for which the

maximum occurs.

84) Objective Function z = 8x + 7y

Constraints 0 ≤ x ≤ 10

0 ≤ y ≤ 5

3x + 2y ≥ 6

84)

Solve the problem.

85) Mrs. White wants to crochet hats and afghans for a church fundraising bazaar. She needs 8

hours to make a hat and 4 hours to make an afghan, and she has no more than 60 hours

available. She has material for no more than 12 items and no more than 10 afghans. The

bazaar will sell the hats for $14 each and the afghans for $8 each. How many of each

should she make to maximize the income for the bazaar? What is the maximum income?

85)

86) Solve the system of equations using matrices. Use Gaussian elimination with

back-substitution. Give only the values of x and z in your answer.

x - y + 3z = 5

5x + z = 2

x + 3y + z = 5

86)

87) Use Gaussian elimination to find the complete solution to the system of equations, or

state that none exists.

x + y + z = 9

2x - 3y + 4z = 7

87)

88) Give the order of the matrix, and identify the given element of the matrix.

-14 8 14 -1

9 -7 -2 2; a12

88)

89) Solve the problem.

Let A = -1 0

4 3 and B = -1 4

3 1. Find A - B.

89)

90) Find the product AB, if possible.

A = 1 3 -1

2 0 3, B =

3 0

-1 1

0 3

90)

91) Find the product AB, if possible.

A = 3 -2 1

0 4 -1, B = 5 0

-2 1

91)

92) Solve the system using the inverse that is given for the coefficient matrix. Give only the

values of x and y in your answer.

x + 2y + 3z = 1

x + y + z = -8

2x + 2y + z = 2

The inverse of 1 2 3

1 1 1

2 2 1

is -1 4 -1

1 -5 2

0 2 -1

.

92)

11

93) Encode or decode the given message, as requested, numbering the letters of the alphabet

1 through 26 in their usual order.

Use the coding matrix A = -1 -3

2 5 to encode the message CARE.

93)

94) Use Cramer's rule to solve the system. Give only the value of y as your answer.

5x + 4y = 4

3x + y = -6

94)

95) Evaluate the determinant.

3 5 1

6 5 6

3 5 3

95)

12