9/18/22, 6:59 PM
Review Test 1-Azime Saydam
Instructor: Azime Saydam
Course: Math 2312 - 03S Precalculus
Mathematics
Student: _____________________
Date: _____________________
Assignment: Review Test 1
1. List all numbers from the given set that​are: a. natural​numbers, b. whole​numbers, c.​integers, d. rational​numbers, e.
irrational​numbers, f. real numbers.
− 13, −
5
6
, 0, 0.3,
7 , π,
25
a. List all the natural numbers from the given set.
​(Use commas to separate answers. Type each number as it​appears.)
b. List all the whole numbers from the given set.
​(Use commas to separate answers. Type each number as it​appears.)
c. List all the integers from the given set.
​(Use commas to separate answers. Type each number as it​appears.)
d. List all the rational numbers from the given set.
​(Use commas to separate answers. Type each number as it​appears.)
e. List all the irrational numbers from the given set.
​(Use commas to separate answers. Type each number as it​appears.)
f. List all the real numbers from the given set.
​(Use commas to separate answers. Type each number as it​appears.)
Answers
25
0, 25
− 13,0, 25
− 13, −
5
6
,0,0.3, 25
7 ,π
− 13, −
5
6
,0,0.3, 7 ,π, 25
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Review Test 1-Azime Saydam
2. Write the set {x | x < − 5} using interval​notation, then graph the set on a number line.
Choose the correct interval notation for the set.
[ − 5,∞)
( − 5,∞)
( − ∞, − 5)
( − ∞, − 5]
Choose the correct graph below.
A.
B.
-10 -8
-6
-4
-2
0
2
4
6
8
10
C.
-10 -8
-6
-4
-2
0
2
4
6
8
10
-10 -8
-6
-4
-2
0
2
4
6
8
10
D.
-10 -8
-6
-4
-2
0
2
4
6
8
10
Answers ( − ∞, − 5)
A. -10 -8 -6 -4 -2 0 2 4 6 8 10
3. Write the set {x | 4 ≤ x < 7} using interval notation and then graph the set on a number line.
The interval notation for the set {x | 4 ≤ x < 7} is
.
Choose the correct graph below.
4
A.
C.
7
0
2
4
6
8
10
0
2
4
6
8
10
B.
D.
0
2
4
6
8
10
0
2
4
6
8
10
Answers [4,7)
D.
0
2
4
6
8
10
4. Using the variable​x, write the interval [ − 5,6] using​set-builder notation.
[ − 5,6] = ​{x ​|
​}
​Answer: − 5 ≤ x ≤ 6
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Review Test 1-Azime Saydam
5. Determine the domain and the range of the​relation, and tell whether the relation is a function.
​{(5​,9​), ​(30​, − 4​), ​(41​,9​), ​(5​,3​), ​(59​,3​)}
The domain is ​{
​}.
​(Use a comma to separate answers as needed. Type each answer only​once.)
The range is ​{
​}.
​(Use a comma to separate answers as needed. Type each answer only​once.)
Is this relation a​function?
Yes
No
Answers 5,30,41,59
9, − 4,3
No
6. Decide whether the relation defined by the graph to the right
defines a​function, and give the domain and range.
10
y
8
Does the graphed relation define a​function?
6
4
No
2
Yes
-10 -8 -6 -4 -2
What is the domain of the graphed​relation?
-2
x
2
4
6
8 10
-4
-6
​(Type your answer in interval notation.)
-8
-10
What is the range of the graphed​relation?
​(Type your answer in interval notation.)
Answers Yes
( − ∞, ∞)
( − ∞, ∞)
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Review Test 1-Azime Saydam
7. Determine the domain D and range R of the following
relation from its​graph, and tell whether the relation is a
function.
The domain is
y
6
4
.
2
​(Type your answer in interval notation.)
x
The range is
.
​(Type your answer in interval notation.)
Is the graph to the right the graph of a​function?
-2
2
4
6
8
-2
-4
-6
Yes
No
Answers [3,∞)
[2,∞)
Yes
8. Evaluate the function for the given value of x.
2
​f(x​) = 5x + 9x − 7​, x = − 4
f​( − 4​) =
​(Simplify your​answer.)
​Answer: 37
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Review Test 1-Azime Saydam
9. Determine the largest open intervals of the domain over
which the function is ​(a)​increasing, ​(b)​decreasing, and ​(c)
constant. Then give the ​(d) domain and ​(e) range.
10
8
6
(​ a) Over what open​interval(s), if​any, is the function​
increasing? Select the correct choice below​and, if​
necessary, fill in the answer box to complete your choice.
4
2
-10 -8 -6 -4 -2
-2
A. The function is increasing on the open​
interval(s)
.
​(Type your answer in interval notation.
Use a comma to separate answers as​
needed.)
y
x
2
4
6
8 10
-4
-6
-8
-10
B. The function is never increasing.
(​ b) Over what open​interval(s), if​any, is the function​
decreasing? Select the correct choice below​and, if​
necessary, fill in the answer box to complete your choice.
A. The function is decreasing on the open​
interval(s)
.
​(Type your answer in interval notation.
Use a comma to separate answers as​
needed.)
B. The function is never decreasing.
(​ c) Over what open​interval(s), if​any, is the function​
constant? Select the correct choice below​and, if​necessary,
fill in the answer box to complete your choice.
A. The function is constant on the open​
interval(s)
.
(​ Type your answer in interval notation.
Use a comma to separate answers as​
needed.)
B. The function is never constant.
​(d) The domain of the function is
.
​(Type your answer in interval notation.)
(​ e) The range of the function is
​(Type your answer in interval notation.)
.
Answers A. The function is increasing on the open​interval(s)
( − ∞,2)
.
​(Type your answer in interval notation. Use a comma to separate answers as​needed.)
A. The function is decreasing on the open​interval(s)
(5,∞)
.
​(Type your answer in interval notation. Use a comma to separate answers as​needed.)
A. The function is constant on the open​interval(s)
(2,5)
.
​(Type your answer in interval notation. Use a comma to separate answers as​needed.)
( − ∞, ∞)
( − ∞,3]
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Review Test 1-Azime Saydam
10. Determine the largest open​interval(s) of the domain over
which the function shown in the graph below is ​(a)​
increasing, ​(b)​decreasing, and ​(c) constant. Then give the
​(d) domain and ​(e) range.
11
10
9
8
7
6
5
4
3
2
1
-1
-8 -7 -6 -5 -4 -3 -2 -1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
(​ a) Over what open​interval(s), if​any, is the function​
increasing? Select the correct choice below​and, if​
necessary, fill in the answer box to complete your choice.
A. The function is increasing on the open​
interval(s)
.
y
(​ Type your answer in interval notation.
Use a comma to separate answers as​
needed.)
x
1 2 3 4 5 6 7 8
B. The function is never increasing.
(​ b) Over what open interval(s), if​any, is the function​
decreasing? Select the correct choice below​and, if​
necessary, fill in the answer box to complete your choice.
A. The function is decreasing on the open​
interval(s)
.
​(Type your answer in interval notation.
Use a comma to separate answers as​
needed.)
B. The function is never decreasing.
(​ c) Over what open​interval(s), if​any, is the function​
constant? Select the correct choice below​and, if​necessary,
fill in the answer box to complete your choice.
A. The function is constant on the open​
interval(s)
.
​(Type your answer in interval notation.
Use a comma to separate answers as​
needed.)
B. The function is never constant.
(​ d) The domain of the function is
​(Type your answer in interval​notation.)
​(e) The range of the function is
.
.
(​ Type your answer in interval notation. Use integers or
decimals for any numbers in the​answer.)
Answers B. The function is never increasing.
A. The function is decreasing on the open​interval(s) ( − ∞, − 3),(4,∞) .
​(Type your answer in interval notation. Use a comma to separate answers as​needed.)
A. The function is constant on the open​interval(s)
( − 3,4)
.
​(Type your answer in interval notation. Use a comma to separate answers as​needed.)
( − ∞, ∞)
( − ∞, − 1.5]∪[ − 1,∞)
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Review Test 1-Azime Saydam
11. Each function is either even or odd. Use ​f( − ​x) to state which situation applies.
4
2
f(x) = x − 6x + 8
The function is (1)
(1)
because (2)
(2)
odd
even
f( − x) = f(x).
f( − x) = − f(x).
Answers (1) even
(2) f( − x) = f(x).
12. Each function is either even or odd. Use ​f( − ​x) to state which situation applies.
3
f(x) = 9x − 3x
The function is (1)
(1)
because (2)
(2)
even
odd
f( − x) = f(x).
f( − x) = − f(x).
Answers (1) odd
(2) f( − x) = − f(x).
13. Determine whether the graph of the given function is symmetric with respect to the​y-axis, symmetric with respect to the​
origin, or neither. Use your calculator and the standard window to support your conclusion.
7
3
​f(x) = x − 5x
Choose the correct answer below.
A. The graph of the function ​f(x) = x7 − 5x3 is symmetric with respect to the y − axis.
B. The graph of the function ​f(x) = x7 − 5x3 is neither symmetric with respect to the y − axis nor
symmetric with respect to the origin.
C. The graph of the function ​f(x) = x7 − 5x3 is symmetric with respect to the origin.
​Answer: C. The graph of the function ​f(x) = x7 − 5x3 is symmetric with respect to the origin.
14. Write the equation that results in the desired translation.
The squaring ​function, shifted 3 units downward.
y=
​(Type an expression using x as the​variable.)
​Answer: x2 − 3
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Review Test 1-Azime Saydam
15. Write an equation that results in the desired translation.
The absolute value​function, shifted to the left 3 units.
Which of the following is the equation of the shifted​function?
A. y = x − 3
B. y = x + 3
C. y = x − 3
D. y = x + 3
​Answer: B. y = x + 3
16. Find an equation that shifts the graph of f by the desired amounts. Graph f and the shifted
graph in the same​xy-plane.
​f(x) = x2 ​, shifted right 3 units and downward 5 units.
Which of the following is the equation of the function of the shifted​graph?
2
y = (x − 3) − 5
2
y = (x + 3) − 5
2
y = (x + 3) + 5
2
y = (x − 3) + 5
Choose the correct graph below that indicates the graph of​f(x) and the shifted graph.
A.
B.
10
y
C.
10
x
-10
y
D.
10
x
10
-10
-10
10
-10
y
10
x
-10
10
-10
y
x
-10
10
-10
Answers y = (x − 3)2 − 5
10
y
x
-10
B.
10
-10
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Review Test 1-Azime Saydam
17. Use transformations of graphs to sketch the graphs of y1 ​,
10
y2 ​, and y3 . Check by graphing in an appropriate viewing
8
window of a calculator.
y1 = x​,
y2 = x + 4,
y
6
y3 = x − 6
4
Use the graphing tool to graph the three functions.
2
x
-10
-8
-6
-4
-2
2
4
6
8
10
-2
-4
-6
-8
-10
​Answer:
10
y
8
6
4
2
x
-10
-8
-6
-4
-2
2
4
6
8
10
-2
-4
-6
-8
-10
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Review Test 1-Azime Saydam
18. The graph of y = ​f(x) has been transformed to the graph of
y = ​g(x) as shown to the right. No shrinking or stretching is
involved. Give the equation of y = ​g(x).
10
y
8
6
​g(x) =
4
​(Simplify your​answer.)
2
-10 -8 -6 -4 -2
-2
-4
2
y = f(x) = x
x
2
4
6
8 10
y = g(x)
-6
-8
-10
​Answer: − (x − 5)2 − 3
19. Fill in each blank with the appropriate response.​(Remember that the vertical stretch or shrink factor is​positive.)
The graph of y = −
1
x − 3 + 8 can be obtained from the graph of y = x by shifting horizontally​___ units to the​___,
5
vertically shrinking by applying a factor of​___, reflecting across the​___-axis, and shifting vertically​___ units in the​___
direction.
The graph of y = −
1
5
the (1)
vertically shrinking by applying a factor of
(2)
(1)
x − 3 + 8 can be obtained from the graph of y = x by shifting horizontally
​-axis, and shifting vertically
left,
(2)
right,
y
(3)
x
units in the (3)
units to
​, reflecting across the
direction.
downward
upward
Answers 3
(1) right,
1
5
(2) x
8
(3) upward
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Review Test 1-Azime Saydam
20. Use the graph of y = ​f(x) shown on the right to choose the graph of y = ​|f(x)| from the four
shown.
10
y
x
-10
10
-10
Choose the graph of y = ​|f(x)|.
A.
B.
10
y
C.
10
y
x
-10
10
y
x
10
-10
-10
10
10
10
y
x
-10
-10
​Answer:
D.
10
x
-10
-10
10
-10
y
x
-10
10
-10
B.
21. The graph of y = ​f(x) is shown. Sketch the graph of y = f(x) .
10
y
x
-10
10
-10
Choose the correct graph of y = f(x) .
A.
B.
10
y
C.
10
x
-10
10
x
10
-10
-10
10
-10
​Answer:
y
D.
10
y
10
x
-10
10
-10
y
x
-10
10
-10
y
x
-10
C.
10
-10
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Review Test 1-Azime Saydam
22. Solve the group of equations and inequalities below analytically.
a. x + 4 = 9
b. x + 4 > 9
c. x + 4 < 9
a. Select the correct answer below​and, if​necessary, fill in the answer box to complete your choice.
A. The solution set is the finite set of numbers
.
​(Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as​
needed.)
B. The solution set is the interval
.
​(Type your answer in interval notation. Type integers or​fractions.)
C. The solution set is ∅.
b. Select the correct answer below​and, if​necessary, fill in the answer box to complete your choice.
A. The solution set is the finite set of numbers
.
​(Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as​
needed.)
B. The solution set is the interval
.
​(Type your answer in interval notation. Type integers or​fractions.)
C. The solution set is ∅.
c. Select the correct answer below​and, if​necessary, fill in the answer box to complete your choice.
A. The solution set is the finite set of numbers
.
​(Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as​
needed.)
B. The solution set is the interval
.
​(Type your answer in interval notation. Type integers or​fractions.)
C. The solution set is ∅.
Answers A. The solution set is the finite set of numbers
− 13,5
.
​(Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as​needed.)
B. The solution set is the interval ( − ∞, − 13)∪(5,∞) .
​(Type your answer in interval notation. Type integers or​fractions.)
B. The solution set is the interval
( − 13,5)
.
​(Type your answer in interval notation. Type integers or​fractions.)
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23. Solve the group of equations and inequalities below analytically.
a. 8 − 2x = 8
b. 8 − 2x ≥ 8
c. 8 − 2x ≤ 8
a. Select the correct answer below​and, if​necessary, fill in the answer box to complete your choice.
A. The solution set is the finite set of numbers
.
​(Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as​
needed.)
B. The solution set is the interval
.
​(Type your answer in interval notation. Type integers or​fractions.)
C. The solution set is ∅.
b. Select the correct answer below​and, if​necessary, fill in the answer box to complete your choice.
A. The solution set is the finite set of numbers
.
​(Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as​
needed.)
B. The solution set is the interval
.
​(Type your answer in interval notation. Type integers or​fractions.)
C. The solution set is ∅.
c. Select the correct answer below​and, if​necessary, fill in the answer box to complete your choice.
A. The solution set is the finite set of numbers
.
​(Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as​
needed.)
B. The solution set is the interval
.
​(Type your answer in interval notation. Type integers or​fractions.)
C. The solution set is ∅.
Answers A. The solution set is the finite set of numbers
0,8
.
​(Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as​needed.)
B. The solution set is the interval ( − ∞,0]∪[8,∞) .
​(Type your answer in interval notation. Type integers or​fractions.)
B. The solution set is the interval
[0,8]
.
​(Type your answer in interval notation. Type integers or​fractions.)
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Review Test 1-Azime Saydam
24. Solve the equation.
​|6x − 2​| + 5 = 8
Select the correct choice below and, if​necessary, fill in the answer box to complete your choice.
A. The solution set is
.
​(Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as​
needed.)
B. The solution is all real numbers.
C. The solution is the empty set.
​Answer: A. The solution set is
−
1 5
,
6 6
.
​(Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as​needed.)
25. Solve the following inequality.
​|12 − ​4x| ≤ 3
Select the correct choice below and, if​necessary, fill in the answer box to complete your choice.
A. The solution set is the interval
.
​(Simplify your answer. Type your answer in interval​notation.)
B. The solution set is the set of real numbers.
C. There is no solution.
​Answer: A. The solution set is the interval
9 15
,
4 4
.
​(Simplify your answer. Type your answer in interval​notation.)
26. Solve the inequality.
The solution set is
.
​(Simplify your answer. Type your answer in interval
notation. Use integers or fractions for any numbers in the​
expression.)
​|1 − 8​x| > 8
​Answer:
− ∞, −
7
8
9
∪
8
,∞
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Review Test 1-Azime Saydam
27. For the piecewise linear​function, find ​(a)​f( − 3​), ​(b)​f( − 1​), ​(c)​f(0), and ​(d)​f(2​).
2x
f(x) =
if x ≤ − 1
x − 1 if x > − 1
​(a)​f( − 3​) =
​(Type an integer or a​decimal.)
​(b)​f( − 1​) =
(Type an integer or a​decimal.)
​(c)​f(0) =
​(Type an integer or a​decimal.)
​(d)​f(2​) =
​(Type an integer or a​decimal.)
Answers − 6
−2
−1
1
28. Graph the​piecewise-defined function.
f(x) =
x − 4 if x ≤ 1
− 3 if x > 1
Choose the correct graph.
A.
B.
10
y
C.
10
x
-10
10
x
10
-10
-10
10
-10
​Answer:
y
D.
10
y
10
x
-10
10
-10
y
x
-10
10
-10
y
x
-10
A.
10
-10
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Review Test 1-Azime Saydam
29. Graph the​piecewise-defined function.
​f(x) =
−3−x
if x ≤ 2
− 6 + 2x
if x > 2
Choose the correct graph below.
A.
B.
5
C.
y
5
y
5
x
-5
D.
5
x
5
-5
5
-5
5
y
x
-5
5
-5
​Answer:
y
x
-5
5
-5
-5
y
x
-5
5
-5
C.
30. Match the following​piecewise-defined function with its graph in choices A to D.
2 if x ≥ 0
h(x) =
− 2 if x < 0
Choose the correct answer below.
A.
B.
4
y
C.
4
y
4
x
-4
y
4
x
4
-4
-4
4
-4
​Answer:
D.
4
y
x
-4
4
-4
x
-4
4
-4
y
x
-4
D.
4
-4
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Review Test 1-Azime Saydam
31. Let ​f(x) = 8x2 and ​g(x) = 4x − 6. Find ​(f + g)(x).
What is the expression ​(f + ​g)(x) equal​to?
A. 8x2 + 4x − 6
B. 8(4x − 6)2
C. 8x2 − 4x + 6
D. 8x2 + 4x + 6
​Answer: A. 8x2 + 4x − 6
32. Let f(x) = x2 − 7 and g(x) = 2 − x. Find (fg)(x).
(fg)(x) =
​(Simplify your​answer.)
​Answer: − x3 + 2x2 + 7x − 14
33. Let ​f(x) = x + 3 and ​g(x) = x2 − x. Find and simplify the expression.
​(f + ​g)(6​)
​(f + ​g)(6​) =
​(Simplify your​answer.)
​Answer: 39
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34.
Review Test 1-Azime Saydam
Let ​f(x) = 4x + 5 and ​g(x) = 2x − 7. Find ​(f + ​g)(x), ​(f − ​g)(x), ​(fg)(x),
​(f + ​g)(x) =
f
g
​(x), ​(f ◦ g​)(x), and ​(g ◦ f​)(x). Give the domain of each.
​(Simplify your​answer.)
The domain of f + g is
.
​(Type your answer in interval​notation.)
​(f − ​g)(x) =
​(Simplify your​answer.)
The domain of f − g is
.
​(Type your answer in interval​notation.)
​(fg)(x) =
​(Simplify your answer. Do not​factor.)
The domain of fg is
.
​(Type your answer in interval​notation.)
f
g
​(x) =
​(Simplify your​answer.)
The domain of
f
is
.
g
​(Type your answer in interval​notation.)
​(f ◦ g​)(x) =
​(Simplify your​answer.)
The domain of f ◦ g is
.
​(Type your answer in interval​notation.)
​(g ◦ f​)(x) =
​(Simplify your​answer.)
The domain of g ◦ f is
.
​(Type your answer in interval​notation.)
Answers 6x − 2
( − ∞, ∞)
2x + 12
( − ∞, ∞)
2
8x − 18x − 35
( − ∞, ∞)
4x + 5
2x − 7
− ∞,
7
2
7
∪
2
,∞
8x − 23
( − ∞, ∞)
8x + 3
( − ∞, ∞)
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Review Test 1-Azime Saydam
35. Use​f(x) and​g(x) to find each composition. Identify its domain.​(Use a calculator if necessary to find the​domain.) ​(a)
​(f ◦ ​g)(x) ​(b)​(g ◦ ​f)(x) ​(c) ​(f ◦ ​f)(x)
3
2
​f(x) = x ​, ​g(x) = x + 3x − 5
(​ a)​(f ◦ ​g)(x) =
​(Type your answer in factored​form.)
The domain of ​(f ◦ ​g)(x) is
​(Type your answer in interval​notation.)
​(b)​(g ◦ ​f)(x) =
​(Simplify your​answer.)
The domain of ​(g ◦ ​f)(x) is
​(Type your answer in interval​notation.)
​(c) ​(f ◦ ​f)(x) =
.
​(Simplify your​answer.)
The domain of ​(f ◦ ​f)(x) is
​(Type your answer in interval​notation.)
Answers
.
2
x + 3x − 5
.
3
( − ∞, ∞)
6
3
x + 3x − 5
( − ∞, ∞)
x
9
( − ∞, ∞)
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Review Test 1-Azime Saydam
36. Use​f(x) and​g(x) to find each composition. Identify its domain.​(Use a calculator if necessary to find the​domain.) ​(a)
​(f ◦ ​g)(x) ​(b)​(g ◦ ​f)(x) ​(c) ​(f ◦ ​f)(x)
2
​f(x) = x ​, ​g(x) =
25 − x
​(a)​(f ◦ ​g)(x) =
​(Simplify your​answer.)
The domain of ​(f ◦ ​g)(x) is
​(Type your answer in interval​notation.)
​(b)​(g ◦ ​f)(x) =
.
​(Simplify your​answer.)
The domain of ​(g ◦ ​f)(x) is
​(Type your answer in interval​notation.)
​(c) ​(f ◦ ​f)(x) =
.
​(Simplify your​answer.)
The domain of ​(f ◦ ​f)(x) is
​(Type your answer in interval​notation.)
.
Answers 25 − x
( − ∞,25]
25 − x
2
[ − 5,5]
x
4
( − ∞, ∞)
37. Let ​f(x) = 2x − 1 and ​g(x) = x2 + 5.
Find ​(f ◦ ​g)( − 7).
Then ​(f ◦ ​g)( − 7​) =
. ​(Simplify your​answer.)
​Answer: 107
38. Subtract.
(−1−8i)−(−2−8i)
(−1−8i)−(−2−8i)=
​(Simplify your answer. Type your answer in the form a​+ bi​.)
​Answer: 1
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Review Test 1-Azime Saydam
39. Multiply.
​(7 + 2i​)(9 − 6i​)
​(7 + 2i​)(9 − 6i​) =
​(Simplify your answer. Type your answer in the form a + bi. Do not​factor.)
​Answer: 75 − 24 i
40. Write the quotient in the form a + bi.
1+7i
1+
i
1+7i
=
1+ i
​(Simplify your answer. Type your answer in the form a + bi. Use integers or fractions for any numbers in the​expression.)
​Answer: 4 + 3 i
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