Final Assessment
College Algebra MATH 107 Spring, 2013, V4.1
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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 [–2.5, 1]; Range [0, 3] B. Domain [–1, 3]; Range [–3, 1] C. Domain [–3, 1]; Range [–1, 3] D. Domain [–3, –1]; Range [0, 1]
2. Solve: 16 6x x− = − 2. ______
A. No solution B. 16/5 C. –8, 2 D. –8
2 4 -4
-2
-4
2
4
College Algebra MATH 107 Spring, 2013, V4.1
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3. Determine the interval(s) on which the function is increasing. 3. ______
A. (–∞, – 4) and (2, ∞)
B. (–6.7, 0) and (3.6, ∞)
C. (– 4, 2) D. (–1, 3)
4. Determine whether the graph of 8y x= − 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: | 4 – 5x | ≥ 6. 5. ______
A. [– 2/5, 2]
B. [– 2/5, ∞)
C. (–∞, – 2/5] ∪ [2, ∞)
D. (–∞, 2] ∪ [–2/5, ∞)
College Algebra MATH 107 Spring, 2013, V4.1
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6. Which of the following represents the graph of 4x + 7y = 28 ? 6. ______ A. B.
C. D.
College Algebra MATH 107 Spring, 2013, V4.1
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7. Write a slope-intercept equation for a line parallel to the line x – 5y = 9 which passes through the point (–10, 7). 7. ______
A. 1
9 5
y x= +
B. 1
5 5
y x= − +
C. 1
7 5
y x= +
D. 5 57y x= +
8. Which of the following best describes the graph? 8. ______
A. It is the graph of a function and it is one-to-one. B. It is the graph of a function but not one-to-one. C. It is not the graph of a function. D. It is the graph of a parabolic relation.
College Algebra MATH 107 Spring, 2013, V4.1
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9. Express as a single logarithm: log (x – 5) + 2 log 1 – 4 log y 9. ______
A. log( 5)
log(4 )
x
y
−
B. 4
5 log
x
y
−
C. 3
log 4
x
y
−
D. ( )log 3 4x y− −
10. Which of the functions corresponds to the graph? 10. ______
A. ( ) 3xf x e−= +
B. ( ) 3xf x e= − +
C. ( ) 4xf x e= +
D. ( ) 4xf x e−= +
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11. A candidate’s share of the votes varies directly as the number of votes cast for the candidate. In a city council election with four candidates, Allison received 1,440 votes and earned a 45% share of the votes. Bell received 832 votes. What was Bell’s corresponding share of the votes? 11. ______ A. 15% B. 26% C. 37% D. 52%
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 – 2) – 6 B. g(x) = f (x + 2) – 6 C. g(x) = f (x – 6) – 2 D. g(x) = f (x + 6) – 2
2 4 -2 -4
-2
-4
2
4
2 4 -2 -4
-2
-4
2
4
College Algebra MATH 107 Spring, 2013, V4.1
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SHORT ANSWER: 13. Multiply and simplify: (6 – i)(7 + 2i). 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 8
x
x
+ ≥
− . Answer: ________
15. A bowl of soup at 185° 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 + 110 e – 0.075 t
Find the temperature of the soup 32 minutes after it is placed in the room. (Round to the nearest degree.) Answer: ________
16. Find the value of the logarithm: 3 1
log 27
. Answer: ________
17. Solve: 7 28 64x − = . Answer: ________
18. Suppose $6,100 is invested in an account at an annual interest rate of 3.7% 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 −−−− 12x + 28.
(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 1 17 1 ( ) 3 ( 3)( 1)( 5)
5 5 5 5 P x x x x x x x= − + + + = − + + −
(a) Which sketch illustrates the end behavior of the polynomial function?
Answer: ________
(b) State the y-intercept. Answer: ________________
(c) State the zeros of the function. 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 18
( ) 5
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 18
( ) 5
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. Simplify: ( )
6 4 3 24 2
5
2a b a b
b
−
−
− . Show work.
23. Points (–8, 1) and (2, 3) 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 of the circle? (c) What is the equation of the circle?
24. Find the equation for a line which passes through the points (–1, 9) and (3, 1). Write the equation in slope-intercept form. Show work.
25. Paul, a resident of Metropolis, pays Metropolis an annual tax of $70 plus 1.8% of his annual income. If Paul paid $1,186 in tax, what was Paul’s income? Show work.
26. Let f (x) = 4x
2 – 9 and g(x) = x – 5.
(a) Find the composite function ))(( xgf � and simplify. Show work.
(b) Find ( ) (3)f g� . Show work.
27. Find the exact solutions and simplify as much as possible: 8x
2 + 17 = 12x. Show work.
28. Given the function 1
( ) 6 7
f x x= − , find a formula for the inverse function. Show work.
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29. The fuel efficiency F for a particular car is given by
F(x) = –0.018 x 2 + 1.476 x + 4.4
where x is the speed of the car in miles per hour (mph) and F(x) is the corresponding fuel efficiency in miles per gallon (mpg). (a) What is the fuel efficiency if the car’s speed is 60 mph? (b) What speed will yield the maximum fuel efficiency? Show work.
30. Solve: 2
12 70 0
5 25
x
x x
+ + =
+ − . Show work.