MATH 107 UMGC Domain and Range of The Piecewise Function Worksheet

Answer multiple algebra questions attached to this assignment. Put the answers on the answer sheet.

MATH 107 FINAL EXAMINATION
This is an open-book exam. You may refer to your text and other course materials as you
work on the exam, and you may use a calculator.
Record your answers and work on the separate answer sheet provided.
There are 30 problems.
Problems #1–12 are Multiple Choice.
Problems #13–21 are Short Answer. (Work not required to be shown)
Problems #22–30 are Short Answer with work required to be shown.
MULTIPLE CHOICE
1. Determine the domain and range of the piecewise function.
1. ______
4
2
-4
-2
2
4
-2
-4
A.
B.
C.
D.
Domain [– 4, 2]; Range [–2, 4]
Domain [– 4, 3]; Range [0, 2]
Domain [–2, 4]; Range [– 4, 2]
Domain [0, 2]; Range [– 4, 3]
2. Solve:
A.
B.
C.
D.
2 x + 20 = x + 6
2. ______
No solution
−14
−8, −2
−2
Page 1 of 11
3. Determine the interval(s) on which the function is increasing.
A.
(–2, 4)
B.
(–1, 3)
C.
(– , –3.6) and (0, 6.7)
D.
(– , –2) and (4, )
3. _______
4. Determine whether the graph of y = x + 2 is symmetric with respect to the origin,
the x-axis, or the y-axis.
4. ______
A.
symmetric with respect to the x-axis only
B.
symmetric with respect to the y-axis only
C. symmetric with respect to the origin only
D. not symmetric with respect to the x-axis, not symmetric with respect to the y-axis, and
not symmetric with respect to the origin
5. Solve, and express the answer in interval notation: | 8 – 5x |
A.
B.
C.
D.
(– , – 4/5]
[– 4/5, 4]
[– 4/5, )
[– 4, 4/5]
12.
5. ______
[5, )
Page 2 of 11
6. Which of the following represents the graph of 3x + 5y = 15 ?
A.
B.
C.
D.
6. ______
Page 3 of 11
7. Write a slope-intercept equation for a line parallel to the line x + 7y = 8 which passes through
the point (–14, 6).
7. ____________
A.
y = − 7 x − 92
B.
y=
C.
1
y = − x+6
7
D.
1
y = − x+4
7
1
x +8
7
8. Which of the following best describes the graph?
A.
B.
C.
D.
8. ______
It is the graph of a parabolic relation.
It is the graph of a function and it is one-to-one.
It is the graph of a function but not one-to-one.
It is not the graph of a function.
Page 4 of 11
9. Express as a single logarithm:
log x – 3 log y + log 1
9. ______
x
y3
A.
log
B.
x +1
log
y
C.
log
D.
log ( x + 1 − 3 y )
3
x +1
3 y
10. Which of the functions corresponds to the graph?
10. ______
A. f ( x ) = −e x + 3
B. f ( x ) = e− x + 2
C. f ( x ) = e x + 2
D. f ( x ) = e x + 3
Page 5 of 11
11. Suppose that for a function f , the equation f (x) = 0 has exactly three real-number solutions.
Which of the following statements MUST be true?
A.
B.
C.
D.
f
f
f
f
11. ______
has exactly three x-intercepts.
has exactly three y-intercepts.
is a polynomial of degree 3.
is an invertible function.
12. The graph of y = f (x) is shown at the left and the graph of y = g(x) is shown at the right. (No
formulas are given.) What is the relationship between g(x) and f (x)?
12. ______
-4
4
4
2
2
2
-2
4
-4
-2
2
-2
-2
-4
-4
y = f (x)
y = g(x)
A.
B.
C.
D.
4
g(x) = f (x – 1) + 2
g(x) = f (x – 2) + 1
g(x) = f (x – 2) – 1
g(x) = f (x + 2) + 1
Page 6 of 11
SHORT ANSWER:
13. Multiply and simplify: (9 + 8i)(5 + 3i).
Write the answer in the form a + bi, where a and b are real numbers.
14. Solve, and write the answer in interval notation:
x −1
x−6
0.
Answer: ________
Answer: ________
15. A bowl of soup at 170 F. is placed in a room of constant temperature of 60 F. The
temperature T of the soup t minutes after it is placed in the room is given by
T(t) = 60 + 110 e – 0.075 t
Find the temperature of the soup 8 minutes after it is placed in the room. (Round to the nearest
degree.)
Answer: ________
16. Find the value of the logarithm: log5
1
.
25
17. Solve: 97 x−3 = 81 .
Answer: ________
Answer: ________
18. Suppose $3,800 is invested in an account at an annual interest rate of 7.4% compounded
continuously. How long (to the nearest tenth of a year) will it take the investment to double in
size?
Answer: ________
19. Let f (x) = x2 − 8x + 15.
(a) Find the vertex.
Answer: ________
(b) State the range of the function.
Answer: ________
(c) On what interval is the function decreasing?
Answer: ________
Page 7 of 11
20. Consider the polynomial P(x), shown in both standard form and factored form.
1
9
19
9
1
P( x) = x 4 + x3 + x 2 − x − 2 = ( x + 5)( x + 4)( x + 1)( x − 1)
10
10
10
10
10
(a) Which sketch illustrates the end behavior of the polynomial function?
A.
B.
vvvv
C.
vvvv
D.
vvvv
vvvv
Answer: ________
(b) State the y-intercept.
Answer: ________________
(c) State the zeros of the function.
Answer: ________________
Answer: ________
(d) State which graph below is the graph of P(x).
GRAPH A
GRAPH B
GRAPH C
GRAPH
Page 8 of 11
21. Let f ( x) =
2− x
.
3+ x
(a) State the domain.
Answer: _________________
(b) State the horizontal asymptote.
Answer: _________________
(c) State the vertical asymptote(s).
Answer: _________________
(d) Which of the following represents the graph of f ( x) =
2− x
?
3+ x
GRAPH A.
GRAPH B.
GRAPH C.
GRAPH D.
Answer: ______________
Page 9 of 11
SHORT ANSWER, with work required to be shown, as indicated.
22. Let f (x) = x – 6 and g(x) =
(a) Find
x+5.
f
(−4) . Show work.
g
(b) Find the domain of the quotient function
f
. Explain.
g
23. Points (5, –10) and (1, 2) are endpoints of the diameter of a circle.
(a) What is the length of the diameter? Give the exact answer, simplified as much as possible.
Show work.
(b) What is the center point C of the circle?
(c) Given the point C you found in part (b), state the point symmetric to C about the x-axis.
24. Find the equation for a line which passes through the points (6, –8) and (3, –2). Write the
equation in slope-intercept form. Show work.
25. A salesperson earns a base salary of $1,850 per month and a commission of 6.4% on the
amount of sales made. If the salesperson has a paycheck of $3,046.80 for one month, what was
the amount of sales for the month? Show work.
26. Let f (x) = 2×2 + 5 and g(x) = x – 3.
(a) Find the composite function ( f g )(x) and simplify. Show work.
(b) Find ( f o g ) (−1) . Show work.
27. Find the exact solutions and simplify as much as possible: 2×2 + 5 = 8x. Show work.
28. Given the function f ( x) = 8 +
1
x , find a formula for the inverse function. Show work.
4
Page 10 of 11
29. Donut Creations, Inc. has determined that when x donuts are made daily, the profit P, in
dollars, is given by
P(x) = –0.002 x2 + 4.3x – 1450
(a) What is the company’s profit if 700 donuts are made daily?
(b) How many donuts should be made daily in order to maximize the company’s profit? Show
work.
30. Solve:
16
x−6
+ 2
= 0 . Show work.
x − 4 x − 16
Page 11 of 11
Problems #1-12 are multiple choice. Record your choice for each problem.
Problems #13-21 are short answer. Record your answer for each problem.
Problems #22-30 are short answer with work required. When requested, show all work and write
all answers in the spaces allotted on the following pages. You may type your work using plaintext formatting or an equation editor, or you may hand-write your work and scan it. In either
case, show work neatly and correctly, following standard mathematical conventions. Each step
should follow clearly and completely from the previous step. If necessary, you may attach extra
pages.
MULTIPLE CHOICE. Record your answer choices.
1.
7.
2.
8.
3.
9.
4.
10.
5.
11.
6.
12.
SHORT ANSWER. Record your answers below.
13.
14.
15.
16.
17.
18.
19.
(a)
(b)
(c)
20.
(a)
(b)
(c)
(d)
21.
(a)
(b)
(c)
(d)
SHORT ANSWER with Work Shown. Record your answers and work.
Problem
Number
Solution
Answers:
(a)
(b)
Work/for part (a) and explanation for part (b):
22
Answers:
(a)
(b)
(c)
Work for part (a):
23
Answer:
Work:
24
Answer:
Work:
25
Answers:
(a)
(b)
Work for part (a) and for part (b):
26
Answer:
Work:
27
Answer:
Work:
28
Answers:
(a)
(b)
Work for (b):
29
Answer:
Work:
30