PreCalculus Final Exam Review
If f ( x)  x 2  1 and g ( x) 
1.
f ( g ( x))
Name________________
1
find each of the following
x
2.
g ( f ( x))
If f ( x)  x 2  2 x  1 and g ( x)  x  2 find each of the following
4. f ( g ( x))
5. g ( f ( x))
Find the zero of each function
7.
f ( x)  x3  3x 2  4 x  12
( f  g )( x)
3.
6.
f ( x) g ( x)
f ( x)  x3  12 x  16
8.
Find the inverse of each function. Graph the function and its inverse.
9.
y  12 x  5
10.
y  ( x  1)2  2
Find the equation of least degree with each of the following roots.
11.
3, 5, 2
12.
4i, -4i, 2
Graph each of the following equations or inequalities
x  1 
x2


1  x  1
13. f ( x)   2 x
 x
x  2 

14.
f ( x)   x 2  6 x
15. y = |x+3| - 2
16.
3x  2 y  4
Identify the domain, intercepts, and asymptotes (horizontal and vertical) of the function.
17.
18.
Simplify each of the following expressions
19. (3x 2 y3 )2 (9 x 2 y 4 )
Solve each equation
21. log x 36  2
22. log5 x  13 log5 64  2log5 3
23. log5 ( x  3)  log5 ( x  1)  1
20. (32 c3d 5 )1/ 5
24. 2.3x  23.4
25. 5 x  2  2 x
Find the value of the six trigonometric functions of an angle  in standard position if a point with the
given coordinates lies on its terminal side.
26. (2, 4)
27. (-4, 1)
Evaluate each expression. Give exact values.
28. cos(cos-1 ¼)
29. cot( Cos 1 23 )
30. cos( Sin 1 53 )
Find each EXACT value.
31.
sin
3
4
33. sec 4π
32. tan
34. cot
State the amplitude (if any), period and phase shift for each function. Then graph each
function.
35. y  cos 3x
36.
37.
y  3 tan( 2 x  2 )
y  12 sin 2 x
Solve each equation for 0˚ ≤ x ≤ 2π
39.
38. sin 2 x  sin x  0
cos 2x  4cos x  3
40.
5cos x 1  3cos 2x
Find the ordered triple or ordered pair that represents the vector CD for each set of given
coordinates. Then find the magnitude.
41.
C(3, 6) D (5, -2)
42.
C(2, 3, -1) D(-3, 0, 5)
43. C(5, 6, -8) D(-2, -4, -8)
Find each of the following if b  1, 3 , c  2, 2 , v  3,1, 1 and w  5, 2,3
44.
45.
a  3b  c
a  b  2c
46.
47.
u  3v  3w
u  4v  2w
Find each dot product or cross product.
48.
4, 2  2,3
49.
3, 4,1  4, 2, 2
50.
5, 2,5  1,0, 3
Write the polar complex number in rectangular form or rectangular number in polar form.
2(cos 2  i sin 2 )
51.
52.
3  3i
53.
Find the conjugate of 5-12i and its modulus.
54.
Find
Graph each of the following polar coordinates or polar equations
r  2
57.
(2, 56 )
55.

  3
58.
(3, 50)
56.
Identify each of the following conic sections. Then write them in standard form and graph the
equation.
59. x 2  10 x  y 2  8 y  20
60. ( x  1)2  2( y  3)2  25
61. 9 y 2  54 y  4 x 2  8 x  41
62. y 2  2 y  5 x  16  0
For
Perform the matrix operations, if possible.
63. CA
65.
64. AB
66.
3AD
|A|
67.
68.
|D|
-3A + 5B
Solve the system of equations using matrices.
69.
Answer each of the following.
70. Find the 20th term in the arithmetic sequence for which a1  3 and d  4
71. Find the sum of the first nine terms of the geometric series 2 + 4 + 8 + …
72. Find the 6th term of the geometric sequence 13 , 154 , 16
75 ,
73. Write out the first 7 terms of the sequence {an} = {2n + 5}
74. Find
for the arithmetic sequence whose initial term
is -4 and common difference is -3.
Find each limit or state that the limit does not exist.
75.
lim
n 
4n  1
3n
76.
lim
n
2
4n
77.
lim
n 
(4n  5)(n  3)
n
Determine whether each series converges or diverges.
78. 2  4  8 16 
79.
18.75 17.5 16.25 15.00 
80.
65  13  2 53  13
25 
For the following situations, give a probability distribution, find the expected value and answer
other questions as needed.
81. When flipping a coin 4 times, how many heads did you get?
82. When I roll two standard six-sided dice, what is the expected value of the sum of the two
numbers?
83. A raffle has 100 tickets and has a first prize of $200, second prize of $100 and third prize of
$50. Prizewinners are drawn, without replacement, in order (1st, 2nd, and 3rd). How much
should the tickets sell for in order for this raffle to be fair?