Math 1050 Final Exam Review
The problems below are intended as a review of the topics that will be covered on the final exam. The final exam will consist of a subset (about 40) questions, which will be drawn from the types of questions in this sheet, with different numbers. You should also be sure to study from the topics list, which is posted separately.
Sets and Numbers
1. Put the following sets in order from smallest to largest:
R
,
Z
,
N
,
Q
.
2. Decide whether each of the following statements is true or false:
(a) π ∈
√
(b) 2
Q
/
R
(c) 0 ∈
Q
(d) 0 ∈
N
(e)
N
⊆
Z
− { 0 }
Rules for Numbers
3. Determine whether the following statements are true or false:
(a) 2 ∈ (2 , ∞ )
(b) π ∈ (3 , 5)
(c) 27 ∈ ( −∞ , 3)
4. Determine whether each statement is true or false. State which rules of numbers (law of associativity, commutativity, inverses, or identity, or the distributive law) are followed or broken. If it is false, write a true statement that follows the rules.
(a) 3 x − 3 y = 3( x + y )
(b) 3
1
3
= 0
(c) x + 2 x = x (1 + 2)
(d) 3 x
2
+ 1 = 3( x
2
+ 1)
5. Write down the following rules for numbers:
(a) Law of Associativity for Addition
(b) Law of Commutativity for Addition
(c) Law of Inverses for Addition
(d) Law of Identity for Addition
(e) Law of Associativity for Multiplication
(f) Law of Commutativity for Multiplication

(g) Law of Inverses for Multiplication
(h) Law of Identity for Multiplication
(i) Distributive Law
6. Write the following set as a set or an interval: “The set of real numbers which are greater than or equal to -3.”
7. Write the following set as a set or an interval: “The set of rational numbers, except for
2 , − 4 , and
1
2
.
Solving Some Simple Equations
8. Solve for x if 3 x − 2 = 4 x − 5, using the rules from the previous section.
Intro to Functions
9. Determine whether the following rule is a function:
“Assign to every real number x the positive real number y such that y 2 = x .”
10. What is the target of the function g :
N
→ [1 , ∞ ) where g ( x ) = 2 x +
1
2
?
11. If f ( x ) =
2 x
2
+1 x − 3
, find f (2). Write your answer in simplest form.
Sequences
12. Decide whether each of the following sequences is arithmetic, geometric, or neither. If it is arithmetic, find the common difference. If it is geometric, find the common ratio.
(a) 3 , − 6 , 12 , − 24 , . . .
(b) 3 , 6 , 9 , 12 , . . .
(c) 115 , 108 , 101 , 94 , . . .
(d) 8 , − 8 , 8 , − 8 , . . .
(e) 1 , 4 , 9 , 16 , . . .
13. Find the 56th term of the arithmetic sequence 12 , 5 , − 2 , − 9 , . . .
.
14. Find the 33rd term in the geometric sequence 4 , 2 , 1 ,
1
2
, . . .
.
Sums and Series
15. Find the sum
3
X
( i
2
+ 1). Write your answer as an integer in simplest form.
i =0
16. Find the sum
100
X
(2 i ). Write your answer as an integer in simplest form.
i =1
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17. Find the sum of the first 100 terms of the arithmetic sequence 4 , − 2 , − 8 , − 14 , . . .
. Write your answer as a single term (you do not need to simplify further).
18. Find the value of the series 6 + 2 +
2
3
(you do not need to simplify further).
+
2
9
+ . . .
and write your answer as a single term
Counting I & II
19. There are 100 questions in a test bank, and a teacher wants to create a practice exam with 50 questions. If the order of the questions doesn’t matter, and the teacher doesn’t want to repeat any questions, how many different exams are possible?
20. Suppose that you are on a scholarship committee at a university. You have chosen the top 10 students, and you have 10 different scholarships to award. How many ways are there to award one scholarship to each of the top students?
21. You are teaching a high school English class. Your school has a list of 20 recommended books, and you have time to read 8 of them over the course of the semester. If the order in which you read the books matters, how many semester-long literary adventures can you create for your students?
22. Write
12
9 as a natural number in simplest form.
23. Use the Binomial Theorem to rewrite ( x − 2 y ) 6 numbers that look like n k
.
so that your answer doesn’t include any
More on Functions
24. If f ( x ) = x 2 + 1 and g ( x ) = 4 x , find f ◦ g ( x ).
25. Find the implied domain of
26. Find the implied domain of f g (
( x x
) =
) =
27. Find the implied domain of h ( x ) =
3 x
2
+1
( x − 2)( x +5)
.
√
4 8 − 4 x .
√
5 x − 3.
28. What is the domain of the function f :
N
→
R
?
29. What is the target of the function
√
2 as defined in class?
30. What is the range of the function f :
R
→
R where f ( x ) = x 2 ?
Intro to Graphs
31. Graph the following functions. Label any x and y -intercepts, and at least one point on the graph which is not an intercept. (You must use numbers to indicate scale. “Tick marks” will not be assumed to have any “usual” meaning.)
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(a) id ( x )
(b) x
2
(c) x 5
(d)
(e) x
1
3
1 x 4
(f) f ( x ) = 4
(g) g : {− 2 , 0 , 3 , 4 } →
R where g ( x ) = x + 2.
32. Is the following graph the graph of a function? Explain why or why not.
33. Given the graph of f ( x ) below, answer the following questions.
(a) What is f (2)?
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(b) What is the domain of f ?
(c) What is the range of f ?
(d) What are the x -intercepts of the graph?
(e) What are the y -intercepts of the graph?
(f) Assuming that the target of f is equal to its range, determine whether f has an inverse function. Explain your reasoning.
Graph Transformations
34. Graph the following functions. Label any x and y intercepts, vertical or horizontal asymptotes.
√
5 (a) 32 x − 2
(b) log
1
3
( − x )
(c) 2 x +3
(d) − 2( x + 1) 2
(e) parabola.)
√
2 − x + 1
(f)
− 1 (In addition to the above instructions, label the vertex of the x 2
1 x n
1
+2 x +1
(Hint: Turn this function into a transformation of a function that looks like
, and then graph.)
35. Given the graph of f ( x ) in question 33, graph the function − f ( x ).
Inverse Functions
36. If f ( x ) = 2 x + 1 and g ( x ) =
1
2 x − 1, are f and g inverse functions?
37. Find the inverse of the function h ( x ) =
3 x − 1
.
x +3
38. The graph of a function f :
R
→ [ − 3 , ∞ ) is shown below. Does f have an inverse?
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39. The graph of a function g :
R
→ (0 , ∞ ) is shown below. Does g have an inverse?
40. The graph of an invertible function h :
R
→ (0 , ∞ ) is shown below, along with the graph of the line y = x . Graph the inverse function h
− 1
( x ).
n th Roots
41. What is the domain of the function
√
4
:
R
→
R
?
**For implied domain questions, see the questions from ”More on Functions.”
42. Graph the function f ( x ) = 2
√
4 x + 1 and label any x and y intercepts.
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43. Graph the function g ( x ) =
√
5 x − 1 and label any x and y intercepts.
Basics of Polynomials
44. What is the leading coefficient of the polynomial 3( x − 2)( x + 2)( x
2
+ 1)?
45. If p ( x ) = 3 x 2 + 4 x + 1 and q ( x ) = x 4 − 2 x + 3, find the polynomial p ( x ) − q ( x ).
46. True or false: The polynomial 4 x 1 7 + 2 x 2 + 3 has 19 roots.
47. What is the degree 4 coefficient of the polynomial 4 x
1
7 + 2 x
2
+ 3?
48. If p ( x ) = 2 x 4 simplest form.
+ 5 and q ( x ) = − x 5 + 3 x , find their product, p ( x ) q ( x ), and write it in
Division
49. Find the quotient:
2 x 4 + 3 x x 2
2 − x + 1
.
− 2
50. Find the quotient
3 x
5
+ 2 x + 5
.
x + 1
Roots and Factors
51. Is 3 x 4 − 3 x 3 + 3 x 2 − 3 x + 5 a factor of the polynomial 3 x 5 + 2 x + 5?
52. Given that − 2 is a root of the polynomial p ( x ) = 3 x
3
+ 2 x
2 product of a linear polynomial and a quadratic polynomial.
− 10 x − 4, write p ( x ) as the
53. Given the graph of the polynomial p ( x ) below, list as many linear factors of p ( x ) as you can.
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54. Is − 3 a root of the polynomial p ( x ) = 2 x 4 − 9 x 2 + 2 x + 6?
55. What is the degree of the polynomial (3 x
1
5 + 6)( x
2
+ 1)( x − 2)?
Constant and Linear Polynomials
56. Graph the function h ( x ) = − 3 and label any x - and y -intercepts.
57. How many roots does the polynomial p ( x ) = 5 have?
58. Graph the polynomial q ( x ) = 2 x − 4 and label any x and y -intercepts.
59. Find all roots of the polynomial p ( x ) = 3 x − 5.
60. What is the slope of the line that passes through the points (5 , − 1) and (2 , 3)?
Quadratic Polynomials
61. Complete the square: Write the polynomial p ( x ) = − 2 x
2 − 4 x +3 in the form α ( x + β )
2
+ γ where α, β, γ ∈
R
.
62. Graph the polynomial q ( x ) = − 2( x − 1) 2 + 4 and label the y -intercept.
63. How many roots does the polynomial − x 2 − 3 x + 1 have?
64. Find all roots of the polynomial x 2 − 2 x − 2.
Factoring Polynomials
65. Is the polynomial (2 x 2 + 1)( x − 2)( x + 3)( x − 1) completely factored?
66. Completely factor the polynomial p ( x ) = 3 x − 1.
67. Completely factor the polynomial q ( x ) = 2 x
2 − 4 x + 1.
68. Find a root of the polynomial p ( x ) = − 2 − x − 3 x 2 − 2 x 3 + 2 x 4 .
69. Completely factor the polynomial 3 x
3 − 5 x
2 − 11 x − 3 given that 3 is a root.
70. Find all roots of the polynomial p ( x ) = − 2( x − 1)( x + 2)( x − 1)( x )( x 2 + 2).
71. List all of the monic linear factors of q ( x ) that you know from its graph:
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Graphing Polynomials
72. Graph the polynomial − 3( x − 2)( x + 4)( x + 1)( x
2
+ 1), and label the x - and y -intercept.
Rational Functions
73. Find all x -intercepts of the rational function r ( x ) =
2( x − 2)( x + 1)( x 2 + 4)
− 2( x + 4)( x 2 + 1)
74. Find all vertical asymptotes of the rational function r ( x ) =
2( x − 2)( x + 1)( x 2 + 4)
− 2( x + 4)( x 2 + 1)
75. Calculate the end behavior of the rational function r ( x ) =
2( x − 2)( x + 1)( x
2
+ 4)
− 2( x + 4)( x 2 + 1)
(In other words, find the function that r ( x ) looks like to the far left and far right of its graph.
76. Graph the rational function r ( x ) =
2( x − 2)( x + 1)( x 2 + 4)
− 2( x + 4)( x 2 + 1)
Exponential Functions
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77. Rewrite the following expression as a single power of a : r
4 a 2 a − 3
78. Graph the function f ( x ) = − (
1
2
) x + 2, and label any x and y -intercepts.
79. Suppose your credit card company charges you 3% interest each month. If you initially owe $5,000, how much will you owe after 1 year (12 months) of not making payments?
80. If
3 x
9
= 3, solve for x .
81. Why can’t an exponential function have base 1?
Logarithms
82. Graph the function g ( x ) = log
3
( x + 3) and label any x and y -intercepts.
83. Find the implied domain of the function h ( x ) = log
1
2
(3 a set or an interval, using the notation from Chapter 1.
− 2 x ) and write your answer as
84. Rewrite the expression log e
( x
2 y 3 z log e
( z ).
) as a sum of terms involving only log e
( x ) , log e
( y ) , and
85. Rewrite the expression log
2
( x ) − 2 log
2
( y ) + 3 log
2
( z ) as a single logarithm.
86. Rewrite log
2
15 in terms of logarithms with base 10.
87. What is the greatest integer which is less than log
5
(59)?
88. Write log
1000
(100) as a rational number. (Compute the exact value of log
1000
(100).
89. If log
4
( x ) = 5
4
, solve for x .
Exponential and Logarithmic Functions
90. Solve for x if log
5
( x + 1) + log
5
(2 x ) = 3 log
5
(2) + 1.
91. Solve for x if 32(4 x ) = 2 x .
Piecewise Defined Functions
92. Find p (1) if p ( x ) =
 x


2
− 2
+ 1 if x ∈ ( −∞ , 1) if x = 1

 x
3 if x ∈ (1 , ∞ )
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93. Graph the function: f ( x ) =

− 2 if x ∈ ( − 2 , 0)


3 if x ∈ [0 , 4)


− 1 if x ∈ (4 , ∞ )
94. The function g ( x ) = | x + 17 | measures the distance from x to what number?
95. Solve for x : | 2 − 4 x | < 5.
96. Compute the exact value of the following expression:
| 2(3) − 4 || − 2 | + 1
97. State the Triangle Inequality, and briefly explain what it means.
Linear Equations in Two Variables
98. Is x = 1 and y = − 1 a solution to the following system of equations?
(
2 x + 2 y = 0
2 x − 2 y = − 4
99. Graph the solution set of the linear equation 3 x − y = − 3.
100. Does the following system of equations have a unique solution?
( x − 2 y = − 1
− 2 x + 4 y = − 2
Substitution
101. The following system of equations has a unique solution. Find the solution:
(
2 x − 3 y = 11
3 x − y = − 1
Linear Equations in Three Variables
102. Is x = 1 , y = − 1 , z = 0 a solution to the following system of equations?

2 x − 3 y + z = 5


3 x + z = 1


− x − 3 y + 6 z = 2
Vectors and Matrices
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103. Compute the following matrices and vectors:
(a)


2
− 1
17
 

+

− 3
0
10


(b)


2 3 − 1
− 1 7 2
0 4 − 5
 
 
− 1
0
2



(c) 2

5
− 1
3


(d)
5 3
0 2
−
0
1


− 1 0
2 0
0 1


104. Are the following matrices inverses?


− 2 0 1
0 − 1 − 1
1 1 0


, and


− 1 − 1 − 1
1 1 2
− 1 − 2 − 2


105. Write down the 2 x 2 identity matrix.
Determinants and Inverse Matrices
106. Explain the relationship between determinants and inverses.
107. Compute det
− 1 2
− 2 1
.

108. Compute det

− 1 0 2
− 2 0 3
1 1 1


109. Does the matrix


− 1 0 2
− 2 0 3
1 1 1

 have an inverse?
110. Find the inverse of the matrix:
− 1 0
− 2 − 3
Matrix Equations
111. Solve the following system of equations
(
3 x − y = 0
− 2 x + y = 2
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112. Use the fact that


− 1 0 2
− 2 0 3
1 1 1

− 1


=

3 − 2 0
− 5 3 1
2 − 1 0

 to solve the system of equations

− x + 2 z = − 1


− 2 x + 3 z = 1

 x + y + z = 0
.
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