Posted: August 7th, 2015

Algebra

College Algebra MATH 107 Summer, 2015, V3.1
Page 1 of 11
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. You must complete the exam individually.
Neither collaboration nor consultation with others is allowed.
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. ______
A. Domain [–2, 4]; Range [–3, 3]
B. Domain [–2, 1]; Range [–2, 4]
C. Domain [–3, 3]; Range [–2, 4]
D. Domain [–3, 0]; Range [0, 4]
2. Solve: 2x − 6 = 3− x 2. ______
A. No solution
B. 3, 5
C. 3
D. 5
-4 2 4
-2
-4
2
4
-2
College Algebra MATH 107 Summer, 2015, V3.1
Page 2 of 11
3. Determine the interval(s) on which the function is increasing. 3. ______
A. (− 4 / 3, 1)
B. (−1,1) and (3,¥)
C. (−¥,−5/ 3) , (0, 2) , and (11/ 3,¥)
D. (−¥,0) and (2,¥)
4. Determine whether the graph of y = x − 6 is symmetric with respect to the origin,
the x-axis, or the y-axis. 4. ______
A. not symmetric with respect to the x-axis, not symmetric with respect to the y-axis, and
not symmetric with respect to the origin
B. symmetric with respect to the origin only
C. symmetric with respect to the x-axis only
D. symmetric with respect to the y-axis only
5. Solve, and express the answer in interval notation: | 6 – 5x | £ 14. 5. ______
A. [–8/5, 4]
B. (–¥, −8/5] È [4, ¥)
C. (–¥, –8/5]
D. [4, ¥)
College Algebra MATH 107 Summer, 2015, V3.1
Page 3 of 11
6. Which of the following represents the graph of 2x − 7y = 14 ? 6. ______
A. B.
C. D.
College Algebra MATH 107 Summer, 2015, V3.1
Page 4 of 11
7. Write a slope-intercept equation for a line parallel to the line x + 8y = 6 which passes through
the point (24, –5). 7. ______
A. y = −8x −197
B.
1
8
8
y = x −
C.
1
5
8
y = − x −
D.
1
2
8
y = − x −
8. Does the graph below represent a function and is it one-to-one? 8. ______
A. It is a function and it is one-to-one.
B. It is a function but not one-to-one.
C. It is not a function but it is one-to-one.
D. It is not a function and it is not one-to-one.
College Algebra MATH 107 Summer, 2015, V3.1
Page 5 of 11
9. Express as a single logarithm: log x – 3 log y + log 1 9. ______
A.
3
1
log
x
y

+ 
 
 
B. 3 log
x
y


 
 
C.
3
1
log
x
y

+ 
 
 
D. log ( x +1− 3y)
10. Which of the functions corresponds to the graph? 10. ______
A. ( ) 1 x f x = −e −
B. ( ) 1 x f x = e −
C. ( ) 1 x f x e− = −
D. ( ) 2 x f x e− = −
College Algebra MATH 107 Summer, 2015, V3.1
Page 6 of 11
11. Suppose that a function f has exactly two x-intercepts.
Which of the following statements MUST be true? 11. ______
A. The graph of f has exactly two points whose x-coordinates are 0.
B. f is an invertible function.
C. f is a quadratic function.
D. The equation f (x) = 0 has exactly two real-number solutions.
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. ______
y = f (x) y = g(x)
A. g(x) = f (x – 1) + 3
B. g(x) = f (x + 1) – 3
C. g(x) = f (x – 3) + 1
D. g(x) = f (x + 3) – 1
2 -4 -2 4
-2
-4
2
4
-4 -2 2 4
-2
-4
2
4
College Algebra MATH 107 Summer, 2015, V3.1
Page 7 of 11
SHORT ANSWER:
13. Multiply and simplify: (7 + i)(8 + 6i).
Write the answer in the form a + bi, where a and b are real numbers. Answer: ________
14. Solve, and write the answer in interval notation:
1
0
5
x
x
+
£
−
. 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: 4
1
log
64


 
 
. Answer: ________
17. Solve: 67x−3 = 36 . Answer: ________
18. Suppose $3,800 is invested in an account at an annual interest rate of 5.2% 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 − 6x + 14.
(a) Find the vertex. Answer: ________
(b) State the range of the function. Answer: ________
(c) On what interval is the function decreasing? Answer: ________
College Algebra MATH 107 Summer, 2015, V3.1
Page 8 of 11
20. Consider the polynomial P(x), shown in both standard form and factored form.
4 3 2 1 3 3 13 1
( ) 6 ( 2)( 1)( 3)( 4)
4 2 4 2 4
P x = x − x + x + x − = x + x − x − x −
(a) Which sketch illustrates the end behavior of the polynomial function?
Answer: ________
(b) State the zeros of the function. Answer: ________________
(c) State the y-intercept. Answer: ________________
(d) State which graph below is the graph of P(x). Answer: ________
GRAPH A GRAPH B
GRAPH C GRAPH D
A.
B. C. D.
vvvv
vvvv
vvvv vvvv
College Algebra MATH 107 Summer, 2015, V3.1
Page 9 of 11
21. Let
2
( )
3
x
f x
x
+
=
−
.
(a) State the domain. Answer: _________________
(b) State the vertical asymptote(s). Answer: _________________
(c) State the horizontal asymptote. Answer: _________________
(d) Which of the following represents the graph of
2
( )
3
x
f x
x
+
=
−
? Answer: ______________
GRAPH A. GRAPH B.
GRAPH C. GRAPH D.
College Algebra MATH 107 Summer, 2015, V3.1
Page 10 of 11
SHORT ANSWER, with work required to be shown, as indicated.
22. Let f (x) = x + 2 and g(x) = x + 5 .
(a) Find ) 1 (−  


 



g
f
. Show work.
(b) Find the domain of the quotient function
g
f
. Explain.
23. Points (–5, 2) and (–1, 8) 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 (4, –3) and (6, –7). Write the
equation in slope-intercept form. Show work.
25. Malia, a resident of Metropolis, pays Metropolis an annual tax of $46 plus 2.4% of her
annual income. If Malia paid $1,558 in tax, what was Malia’s income? Show work.
26. Let f (x) = 2×2 + 7 and g(x) = x + 6.
(a) Find the composite function ( f o g)(x) and simplify. Show work.
(b) Find ( f o g ) (−5) . Show work.
27. Find the exact solutions and simplify as much as possible: 5×2 + 6 = 12x. Show work.
28. Given the function
1
( ) 5
7
f x = − x , find a formula for the inverse function. Show work.
College Algebra MATH 107 Summer, 2015, V3.1
Page 11 of 11
29. Chocolate Confections, Inc. has determined that when x chocolate cakes are made daily, the
average cost per chocolate cake (in dollars) is given by
C(x) = 0.001 x2 – 0.14x + 15.20
(a) What is the average cost per cake if 40 chocolate cakes are made daily?
(b) How many chocolate cakes should be made daily in order to minimize the average cost per
cake? Show work.
30. Solve:
2
10 48
0
4 16
x
x x
+
+ =
+ −
. Show work.