MATH 0310 Review for Final Exam

1. For the following graph of f ( x) ,
determine:
y


a.
f  1
b.
the domain of f ( x)
1a_________________


c.
x
    

the range of f ( x)





1b_________________

1c_________________



x4
x 5
2.
Use set-builder notation to write the domain for f ( x) 
3.
Use interval notation to write the domain for f ( x)  3 x 2  2 x  1
4.
Determine whether each relation is a function. Answer yes or no.
4a____________
4b ___________
a

y










    

5.
C

y
3 ________________
4c __________
b

2 ________________




x


    





x


    













y
x





Factor completely:
a.
5 x3  40 x 2  45 x
5a_______________
b.
8 x 2  18
5b_______________
c.
6 x 2  11x  10
5c_______________
Rev: Apr. 10, 2013
Page 1
687320415
6. Determine whether each of the following is a function. Answer yes or no.
a)
 2,5 , 1, 6  , 8, 4  ,  2,3 ,  (5, 7 
6a _________________
b)
1,5 ,  2, 4  ,  3,5 ,  4, 6 
6b _________________
c)
from this table of values
6c _________________
x
y
-2
-8
-1
-3
7
2
1
y2
7. Simplify: a.
1
3
y
1
7
2
12
3
-3
1
1

b. x  3 x  3
1
1

x 3 x 3
9
7a________________
7b ________________
2m  11m  21
4m 2  9
2
8. Simplify:
2m  7
2m  3
A)
B)
8 _________________
m7
2m  3
C)
m7
2m  3
D)
2m  7
2m  3
E)
m7
2m  3
For problems 9 - 11, perform the indicated operation and simplify the answer.
9.
2a 2  7ab  15b 2 2a 2  3ab
 2
2ab  10b 2
4a  9b 2
9 _________________
10.
2 x2  5x  7 5x2  5x
 2
4 x2  9
2 x  3x
10 _________________
11.
5
3

x  5 x  6 4 x  12
11__________________
12.
5
8
 2
x  3 x  4 x  16
12__________________
13.
2
2
Solve:
x
18
3
 2

x  4 x  2x  8 x  2
A) x  4
Rev: Apr. 10, 2013
B)
x  2, 4
13_________________
C) x  3
Page 2
D) x  4
E) x  3, 2
687320415
For problems 14 – 17, find the following, given that f x   3x  6 and g x   x 2  1 .
14. g  f x 
14________________
15.  f  g 2
15________________
16. ( f  g )( x)
16______________
g
17. the domain of  x 
f 
17________________
Solve #18, #19 and #20.
18. 2 x  4  0
18________________
19. x  2  5  11
A) x  14, 18
20.
19 ________________
B) x  18
D) x  4
C) x  14,18
E) x  18
x7 9  2
20 ________________
For problems 21 – 23, solve and graph the solution set on a number line. Then write the
solution set in a) interval notation and b) set-builder notation.
21.
2 x  5  5x  4
21a________________
21b ________________
22.
8  5x  18
22a________________
22b ________________
23.
7  x  2  4
23a________________
23b ________________
24. Let f ( x)  5  2 x . Find:
a. f  10 
b. f  0 
d. State the domain of f ( x)
Rev: Apr. 10, 2013
c. f  3
24a_______________
24b_______________
24c_______________
24d_______________
Page 3
687320415
For problems 25 – 28, assume x is nonnegative and simplify.
25.
 27 x 6 y 8
3
25________________
A. 9x 2  y 2  3 y 2 
26.
27.
28.
B. 3x 2 y 2
C. 3x3 y 5
D. 3x 2 y 2  3 y 2 
4 x10
9
 18

3
E. 3x3 y 4
26________________
27_______________
8
8x
125 x 2
28________________
For problems 29 – 35, perform the indicated operations and simplify.
29. 7 x 8 x  2 x 50 x
30.
27 p  75 p
A)
102 p 2
29_________________
30 ________________
B)
p 102
C) 3 p 8
D) 8 3p
E) 8 p 3
31.
3(5 2  6)
32. ( 3  5) 2
31 ________________
32 ________________
33. a. (9  4i )  (1  6i)
33a________________
b. (9  4i)(1  6i)
34. a. Rewrite
33b________________
x 5 with rational exponents.
6
34a________________
1
2
b. Rewrite x in radical notation.
34b________________
35. Rationalize each denominator:
x
a.
3
4
b.
3 2
7
c.
3i
35a________________
35b________________
35c________________
Solve #36 and #37.
36.
37.
x  5 x  11  1
3
36 ________________
3x  1  4
Rev: Apr. 10, 2013
37________________
Page 4
687320415
Solve #38 – #41, using the indicated method.
x 2  3x  18  0
38.
factoring:
38_______________
39.
square root property:
40.
completing the square:
41.
the quadratic formula: 2 x 2  2 x  5  0
x  32
 49
39_______________
x2  4x  4
40_______________
41______________
1 3
 i
D. 1 11 i
2 2
42. A ball is thrown upwards according to the following equation,
h(t)= 16t 2  80t . When will it reach a height of 64 feet?
A. 13i
B. 2, 1
C.
E. 1 11
42_________________
43. Normally, it takes Professor Thompson 3 hours to grade a class of final exam essays.
If his graduate assistant grades the essays, it takes her 5 hours. How long will it take
them to grade the essays together?
43________________
44. A biker can travel 18 mph with no wind. The same rider can
bicycle 8 miles against the wind in the same time it takes to bicycle
12 miles with the wind. What is the speed of the wind?
A) 9 mph
B) 3.6 mph
C) none of these
45. Solve 3 x  5 x  1  0 using a calculator.
Round the answer to three decimal places.
44_________________
D) 90 mph
E) 9 mph
2
45________________
46. When a car comes to a sudden stop, you can determine the skidding
distance (in feet) for a given speed (in miles per hour) by using the formula
d  2 5 x , in which d is skidding distance and x is speed. Calculate the following,
rounding the answers to the nearest tenth:
the skidding distance for a speed of 50 mph
46a _______________
b) the speed when the skidding distance is 20 feet.
46b _______________
a)
47. Solve, writing the solution in the indicated notation
a) set-builder notation:
b)
interval notation:
Rev: Apr. 10, 2013
x 5  9
47a _______________
x  7  19
47b ________________
Page 5
687320415
48. For f  x    x 2  2 x  3 :


a. find the vertex
48b______________


b. find the axis of symmetry
c. find the direction the
parabola opens
48a______________
y

x
    






48c______________

48d______________


d. find the y - intercept
48e______________

48f______________
e. find the x - intercepts
f. find the minimum or
maximum value
48g _____________
g. graph the function.
49.
Graph
y  5 x 2
49________________

y




    

x









50. For the following graph of function f ( x) ,
a)
find the domain b) find the range
Rev: Apr. 10, 2013
50a _______________
50b _______________
Page 6
687320415
MATH 0310 FINAL EXAM REVIEW
**ANSWER KEY**
1. a) 1
 x x  4 or  4,  
b)
26.
 y y  1 or 1,  
c)
27.
2 x5
3
3 2i
28.

3. (, )
2 x2
5
29. 4 x 2 x
4. a) yes b) no c) yes
30.
D)
31.
5 6 3 2
2.
 x x  5
5 x( x  9)( x  1)
5. a)
28  10 3
33. a) 8 10i
2(2 x  3)(2 x  3)
c) (3 x  2)(2 x  5)
b)
6. a) no
b) yes
32.
c) yes
3y 1
x
b)
3
y
m7
8. E)
2m  3
a
9.
2b
2x  7
10.
5(2 x  3)
3x  26
11.
4( x  2)( x  3)
3
12.
( x  1)( x  4)
7. a)
13.
x  3
C)
34.
a)
35.
a)
36.
x5
37.
x  21
38.
x  6, 3
39.
x  3  7i
15. 5
16. 3 x  6 x  3 x  6
2
 x x  2 or (, 2)
(2, )
44.
18. x  2
19. C) x  14,18
45.
46.
20.
47.
no solution
21, 22, and 23—see graphing solutions on next page
24. a) 5
d)
25.

x

5 c) i
5 
5
x   or  , 
2 
2
b)
D) 3x y
2
23
Rev: Apr. 10, 2013
b)
33  50i
x
b)
3x
12  4 2
b)
3
7
21 7
 i
c)
10 10
x  2  2 2
1 3
41.
C) x   i
2 2
42. t  1 second or t  4 seconds
7
43. 1 hours
8
2
17.
x
5
6
40.
14. x  3 x  7
3
8 3p
B) 3.6 mph
x  .232
or x  1.434
a ) 31.6 feet b) 20 mph
a)
 x 4  x  14
b)
 , 12   26, 
48,49--see graphing solutions on next page
50.
a)
3, 
b)
 2,  
y2
Page 7
687320415
SOLUTIONS FOR GRAPHING PROBLEMS
21.
a)
3, 
22.
a)
 , 2
23.
a)
 x x  3
b)
 x x  2
b)

 5, 6

b) x 5  x  6
)
-5
6
48. a) (1,4) b) x=1 c) downward
d) (0,3)
g) graph below
e) (-1,0) and (3,0)
f) maximum is 4
y




x














49.
Rev: Apr. 10, 2013
Page 8
687320415