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COLLEGE ALGEB RAFINAL EXAM REVIEW SHEET
Perform the indicated operation and give a simplified answer:
222
65 25 m -4 5m +17 m +6 7 3x x -9
1. -2. 3. x +4. x +3 x -1 5m -
2 6m 2 +18 mx 4x -12 15 x 7
Simplify the complex rational expressions:
3 1
+
3
y x
5.6.15 9x y
--
2
yyyx
Solve the equations:
7. 2(m -5) =m +1 8. x 2 -6x -
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11 =0 9. 3x 2 +2x =52332
10.3x -5x -1 =0 11. 3x =75 x 12. 12 x +16 x -28 x =013.x 4 -11 x 2 +28 =0 14. 3 19 x +1 -7 =-5 15. x +
2 +3 =x -12x 2 x 2 +10 -20 516. +=17. =x -1x -
3 x 2 -4x +3 x 2 -4 x +2
Solve the inequalities; write your answer using inequality notation:
3 p +4
18.3x +
7 =x -1 19. 4 >>-5-2
Given either the two points that lie on a line or the linear equation, find theslope:
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20.(3,-5) and (-5,7) 21. y =3x -7 22. -5x =-3023.6x -7 y =15 24. y =8 25. 3(x -5) +4 y =45 +4yFind the x and y intercepts:
26. 4x -3y =6 27. x =328. y =-1
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Find the equation of the line that passes through the given points, in slope-intercept form:
29. (30,1) and (3,10) 30. (1,6) and (1,-5) 31. (-1,-7) and (0,-17)32. (8,-2) and (0,-18)Find the equation of the line that passes through the given point, and is parallel to the line whoseequation is given:
33. 7x + 2y = 40; (6, 5) 34. -4x 5y = 20; (-5, 0) 35. y = 2x 4; (3, 4)Find the equation of the line that passes through the given point, and is perpendicular to the line whoseequation is given:
1
36. y = -3x 1; (-4, 2) 37. x 3y = 9; (2, 4) 38. y =x +9 ; (5, -5)6
Sketch the graph of the following lines by plotting the intercepts:
39. y = 2x + 1 40. 4x 3y = 12 41. x 2y = -2 42. 3x + 2y = 6Sketch the graph of the line with slope m that contains point P:
2
43. m =P(1, -4) 44. m = -5 P(-3, 2) 45. m = 0 P(0,3)3Find the following using k(x) =3x 2 -7 :
k(x +
h) -k(x)
46. k(2) 47. k(x +h) 48.h
Find the domain of the function, write your answers in interval notation:
4 -x
49. f (x) =3x 5 -7x +92 50. g(x) =x -7 51. h(x) =52. k(x) =x +6 -5x
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2xx +1 34x +7
53. m(x) =54. f (x) =x -5 55. y =
x 2 -25 6
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Given f(x)and g(x), find the value of the given combination function:
56. f (x) =x -3, g(x) =x +5;( f +g)( -2) 57. f (x) =-4x 2 -2, g(x) =x +7;( f -g)( -5).
f
..
58. f (x) =x -2, g(x) =-4x 2 +20 x +1;( fg )(3) 59. f (x) =4x -3, g(x) =2x 2 +14 x +3;
(-5)
g
Given f(x) and g(x), find the following composite function:
4 5
60. f (x) =3x +14, g(x) =5x -1;( f og)( x) 61. f (x) =, g(x) =;( f og)( x)x +
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3 6x
62. f (x) =4x +4, g(x) =2x 2 +5;(g of )( x) 63. f (x) =2x +4, g(x) =4x 2 +3;(g of )( x)Determine whether the two functions are inverses of each other:
x +7 2 +x 2 33
64. f (x) =7x -8, g(x) =65. f (x) =, g(x) =
66. f (x) =x -5, g(x) =x +5
8 x x -1
Find the inverse of each function:
5
x +2
67. h(x) =2x 5 +3 68. f (x) =69. g(x) =3 x +7 -4 70. k(x) =11 x -6 x +9
For the given quadratic functions, find the vertex and the x-and y-intercepts:
2 22
71. y = x +3 72. f (x) =4(x -1) -7 73. f (x) =-(x -
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5)74. f (x) =x 2 +6x -7 75. f (x) =x 2 -4x +3
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For #76-78, find the x and y intercepts of the polynomial function; write them as ordered pairs. Then,determine the correct graph of the function:
76. f (x) =x(2 -x)(4 -x)A. B. C. D.77. s(x) =x 3 +5x 2 +6x78. r(x) =x 2 -5x -6Use long division to find the quotient and remainder:
2x 3 -4x 2 +7x -3
79.x 2 +3
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Use synthetic division to find the quotient and remainder:
3 54
2x +5x -1 2x -3x +3x
80. 81.x +3 x -2
Use synthetic division to find all remaining zeros of f(x),given that c is a zero of f(x):
42 32
82. f (x) =x -
21 x -100;c =5 83. f (x) =x +2x -5x -6;c =284. f (x) =2x 3 -13 x 2 +10 x +25;c =
-1Given f(x),find the vertical asymptotes:
6 6x -5 4x 3 -x 2 +5x -1
85. f (x) =86. f (x) =87. f (x) =
x +4 10 -2x x 2 -2x -3
Given f(x),find the horizontal asymptotes:
6x -
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5 x x 2 -5x +13
88. f (x) =89. f (x) =290. r(x) =10 -2x x -9 x -2
Given f(x),find the slant asymptotes:
32 2
4x -x +5x -1 x -5x +13
91. f (x) =92. f (x) =x 2 -2x -3 x -2
Solve the polynomial inequalities; write your answer using interval notation.Show your work using a sign chart:
93. (x +1)( x -
3)( x -6) >0 94. x 2 (5 -x) =0 95. 2x 2 =15 x -27Solve the rational inequalities; write your answer using interval notation.Show your work using a sign chart:
x -7 10 4 3
96. =0 97. >5 98. =x +3 x x -1 x -5
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Solve the exponential equations:3x-5 x-2 7x 3x-5 x-2 7x
1
99. 8 =64 100. 100 =10
x x-1
101. 5 =26 102. 7 =13 (exact answer and four decimal places)3x-7
103. 2e =12 (exact answer and four decimal places)104. The population of a certain city is estimated to be increasing according tothe model0.012 tP(t) =300,000 e where t is the number of years from the present. Rounding to the neare
st person,what does the model predict that the population will be in 5 years?
Find the value of each logarithm without using a calculator:1
5 1
105. log 125 106. log 107. log 9 108. log5 23
16 1000
109. log 7 1 110. ln e 4Use the properties of logarithms to expand the expression. Simplify any log expressions that can beevaluated without the use of a calculator:
3 5
.
x pq
7 .
111. log10 x 112. ln.
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e
.7..x 2 y .
113. ln ..114. log125..+250.+125
3
.
.
Use the properties of logarithms to condense the expression as much as possible,writing the answer as asingle log term with a coefficient of 1 and a simplified argument.
115. log 5 x -2log 5 y 116. ln( x 2 y 5 ) -2ln x3
117. ln 3 +ln p -
6ln q 118. 5log (a b)+6log (a)-7log b119. log 7 (x 2 -9) -log 7 (x -3)
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Solve each logarithmic equation; give a decimal approximation when requested:
120. log 2 5x =4 121.log x 3log 2 = 2
122. log 4 8x =log 4 40 123.log(6x +4) =2
124. log 163x =7 125.log (4x +7)-log (x +1)=3
4
22
126.log 3 (x +1) +log 3 (x +3) =1 127. ln( x -2) +ln( x -5) =ln( x -1)
(round to 4 decimal places)
128. log 1 3x =2
3
Solve the system of equations:
7x -3y =26x +
5y =-1 9x -6 y =2
129.130. 131.
3y =13
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5x -2 y =-7 6x -4 y =5
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ANSWER KEY
1. .2. 6.3.4...
6.7. .=118. .=3259. .=-.
.,
.=1. .=..
11. .
=0,.=-5,.=512. .=0,.=
-..,.=113. .=-7,
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.=7,.=-2,.=214. =.. .=7
16. .=6,.=-217. No Solution18. $-4,
8.19. -4,2.. .=-..21. .=322. .=
undefined23. .=..24. .=0. .=undefined26. x-intercept:y-intercept:
-227. x-intercept: 3y-intercept: none28. x-intercept: noney-intercept: -129. .=-.
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.
.+11. .=131. .=-10.-1732. .=2.-1833. .=-...+
2634. .=-...-4. .=2.-
236. .=...+.37. .=-3.+10
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38. .=-6.+2539.40.41.
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42.43.44.
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45.46. 02.=547. 0.+1.=3..+6 .+31.-7232.
48. =6.+3.
49. -8,8.50. $7,8.51. -8,0..0,8.52. 5-8,..53. -8,
-5..-5,5..5,8.54. -8,8.55. -8,8.56. -257. -10458. 25
59.60. 15.+1161.7.
62. 32..
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+64.+3763. 16..+64.+6764. not inverses65. inverses66. inverses98.
67. 1. .=68. :. .=69. ;. .
=.+4.-770. 0. .=71. vertex: 0,3.x-intercepts: noney-intercept: 3
72. vertex: 1,-7.
.
.
x-intercepts: ,y-intercept: -3..
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73. vertex: 5,0.x-intercepts: 5y-intercept: -2574. vertex: -3,-16.x-intercepts: -7,1y-intercept: -775. vertex: 2,-1.x-intercepts: 1,3y-intercept: 376. x-intercepts: 0,2,4y-intercept: 0Graph: A77. x-intercepts: -3,-2,0y-intercept: 0
Graph: C78. x-intercepts: -1,6y-intercept: -6Graph: D79. Quotient: 2.-4Remainder: .+980. Quotient: 2..
-6.+23Remainder: -7081. Quotient: 2..+..+2..
+4.+11Remainder: 2282. .=-583. .=
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-3,.=-184. .=..,.=585. .=486. .=587. .=-1,.=3
88. .=-389. .=090. none91. .=4.+792. .
=.-393. -1,3..6,8.94. $5,8.95. =3,
#.96. -3,7.97. 0,2.98. 1,5..$17,
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8.99. .=1100. =-.101. .=?@.
102. .=?@[email protected]
?@..
103. .=?..2.9306
104. 318,551people105. 3
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106. -4107.108. -3109. 0110. 4111. 1+7log.112. ln.+ln.+5ln.-7113. 2ln
.+ln.-ln3114. 3+2log.+1.
115. log.5..116. ln-.117. ln5.118. log5K.119. log.+3.120. .
=.121. .=800122. .=5123. .=
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16124. .=..125. .=-..126. .=0127. .=4+56.2361128. .=-2129. 5.
.130. -1,1.131. No Solution
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