College Algebra MATH 107 Summer, 2015, V1.4
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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 [–1, 2]; Range [0, 2] C. Domain [–1, 4]; Range [–2, 2] D. Domain [–3, 3]; Range [–2, 4]
2. Solve: 10 3x x− = − 2. ______
A. No solution B. 5 C. –5, 2 D. –5
-2 2 4 -4
-2
-4
2
4
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3. Determine the interval(s) on which the function is increasing. 3. ______
A. (–∞, –1)
B. (–∞, – 3) and (1, ∞) C. (– 2, 2)
D. (– 4.5, – 1) and (2.5, ∞)
4. Determine whether the graph of 3y x= + 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 x-axis only C. symmetric with respect to the y-axis only D. symmetric with respect to the origin only
5. Solve, and express the answer in interval notation: 7 4 5x− ≥ . 5. ______
A. [1/2, 3]
B. (–∞, 3] ∪ [1/2, ∞)
C. (–∞, 1/2] ∪ [3, ∞)
D. [1/2, ∞)
College Algebra MATH 107 Summer, 2015, V1.4
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6. Which of the following represents the graph of 4x + 9y = 36 ? 6. ______ A. B.
C. D.
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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. 7 92y x= − −
B. 1
8 7
y x= +
C. 1
4 7
y x= − +
D. 1
6 7
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 and it is not one-to-one. D. It is not a function but it is one-to-one.
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9. Express as a single logarithm: 2 log (x + 3) – 4 log y + log 1 9. ______
A. 6 log( )
log(4 )
x
y
B. 2
4
9 log
x
y
+
C. ( )
2
4
3 log
x
y
+
D. ( )log 2 7 4x y+ −
10. Which of the functions corresponds to the graph? 10. ______
A. ( ) 2 1xf x e−= −
B. ( ) 1xf x e−= +
C. ( ) 2xf x e− +=
D. ( ) 2xf x e += −
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11. Suppose that for a function f , the equation f (x) = 0 has exactly one real-number solution. Which of the following statements MUST be true? 11. ______ A. f has exactly one real-number zero. B. f is a linear function.
C. f (x) ≥ 0 for all x in the domain of f. D. f 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. ______
y = f (x) y = g(x) A. g(x) = f (x – 3) – 1 B. g(x) = f (x – 1) – 3 C. g(x) = f (x + 1) – 3 D. g(x) = f (x – 1) – 1
2 4 -2 -4
-2
-4
2
4
2 4 -2 -4
-2
-4
2
4
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SHORT ANSWER:
13. Multiply and simplify: (6 + 5i)(8 + i). 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: 4
0 2
x
x
+ ≥
− . Answer: ________
15. A bowl of soup at 165° F. is placed in a room of constant temperature of 75° F. The temperature T of the soup t minutes after it is placed in the room is given by
T(t) = 75 + 90 e – 0.075 t
Find the temperature of the soup 20 minutes after it is placed in the room. (Round to the nearest degree.) Answer: ________
16. Find the value of the logarithm: 3 1
log 9
. Answer: ________
17. Solve: 5 87 49x + = . Answer: ________
18. Suppose $2,400 is invested in an account at an annual interest rate of 7.1% 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) = x 2 −−−− 4x + 7.
(a) Find the vertex. Answer: ________ (b) State the range of the function. Answer: ________ (c) On what interval is the function decreasing? Answer: ________
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20. Consider the polynomial P(x), shown in both standard form and factored form.
3 21 7 7 1 ( ) 5 ( 1)( 3)( 5)
3 3 3 3 P 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
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21. Let 2
2
3 3 ( )
4
x f x
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 2
2
3 3 ( )
4
x f x
x
− =
− ? Answer: ______________
GRAPH A. GRAPH B.
GRAPH C. GRAPH D.
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SHORT ANSWER, with work required to be shown, as indicated.
22. Let f (x) = x + 6 and xxg −= 3)( .
(a) Find )1(−
g
f . Show work.
(b) Find the domain of the quotient function g
f . Explain.
23. Points (–3, 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 (–6, 1) and (–8, 9). Write the equation in slope-intercept form. Show work.
25. A salesperson earns a base salary of $1,650 per month and a commission of 5.3% on the amount of sales made. If the salesperson has a paycheck of $3425.50 for one month, what was the amount of sales for the month? Show work.
26. Let f (x) = 5x
2 – 8 and g(x) = x – 2.
(a) Find the composite function ))(( xgf o and simplify. Show work.
(b) Find ( ) ( 1)f g −o . Show work.
27. Find the exact solutions and simplify as much as possible: 2x 2 + 3 = 6x. Show work.
28. Given the function 1
( ) 7 3
f x x= + , find a formula for the inverse function. Show work.
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29. Donut Delights, Inc. has determined that when x donuts are made daily, the profit P (in dollars) is given by
P(x) = –0.002 x 2 + 4.1x – 1350
(a) What is the company’s profit if 800 donuts are made daily? (b) How many donuts should be made daily in order to maximize the company’s profit? Show work.
30. Solve: 2
5 28
7 49
x
x x
+ =
+ − . Show work.
