1 Fundamental Concepts From Algebra & Precalculus
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1 Fundamental Concepts From Algebra & Precalculus. Review Exercises.. Simplify eac expression.. 5 7) [ 5)) ]. ) 5) 7) [ 5) 8 6)] [ ] ) 8) [ )6)] 7) ) 6 8 ) ) 7) x 5x 5 6x 0. ) 5x 9 5x 5x ). 8x 6 y ) x 5 6 y ) 6. x y ) x 5 6 y ).. x y 7 y 5 8x y 6 ) ) x y y ) 5. xy y 7x ) 6. x y 5 )9x 6 y) / x y / 7. x y ) xy ) 8. 7x 6 y ) 8 x y ) x6 5 y 5 ) 5.. Factor te expression completely and simplify). 9. 6y + y 0. 9x y + 6xy. x. y. 8a b 6. x z 7 6. x + x x + 5x u u y 6y + y 0. x x + 6x. x x x 7x +. 6x 7xy 5y. 6x 9y 5. 6x 6x x 5x 7. x + x x 8. 5 x + 5x 0x 0 9. x + x 5x y y. 6x y 7 y x + 9. x x ) xx ) 5x + 5. x x + ))x + ) ) + x + ) x ). 6x 5) )x + )x) + x + ) )6x 5) 6)
2 5. 5x + ) x)x ) + x + ) 5 )x ) 6. x ) )9 x ) 5 + x ) 5)9 x ) 6x ) 7. x ) )x + ) / + x ) )x + ) / 8. x + ) / x x + ) / 9. x x ) + xx ) 6x x + y) 00x + y) 5. x 7x x 5x x +.. Complete te Square witout canging te value of te expression). 5. x + 6x 5. t 9t 55. y 8y x + x x x x + x Perform te Indicated Operations and completely simplify) x + 6. x x x x 0 8 y 0y 6 5y 6. x x 9 + xx ) 6. x x x 0 + x x x x + x + x x 8 x ) x x + x x + ) ) + ) x + x x + x 70. x + x + x x + 7. r s + s r r s s r 7. x + x + + x x + 7. x x x + 6 x x x 9 x x + 9x 6x + x x 5x + x 75. x 6 x + x 0 x + 0x + x + 8x x 5x x x x 6x x + x 7x x + ) x + ) x x) 78. x x + ) 79. x + ) )x 5)) + x 5) )x + ) ) 80. x + ) 6 )x 5) ) + 6x + ) 5 )x 5) 8. x + 9) )x + 6) + x + 9) x)x + 6) 8. x)x ) x )x ) x) [ x ) ] x x 5) x )x 5) x) [ x 5) ] 8. x x ) x ) x ) x) [ x ) ] 86. x + ) x )x + ) x) [ x + ) ] )x + ) )x + ) x + ) )x + ) ) [ x + ) ]
3 ..5 Rationalize te Numerator and completely simplify). 87. x y x y 88. x + ) + x +..6 Solve te Equation. 89. x x = x + 5x = 9. x = x ) 9. x + x 0) = 5x ) 9. x 6x + = 9. x + )x + ) = 95. x = 5x + 8x 96. x + 8x = 9x 97. x x = x + 8x + = x x + = x + ) 5 x ) = 5 0. x )x + ) = 0. x ) x + ) = 0. z z + ) z + ) = 0 0. x + x x x = x + = 6 + x x + 7 = x + x x = 5x + x S = x 5 x = x + 5 6t t + t = t + p q + p q) for q. f = p + q for q. R = + R + R + R for R. 0 8x = x. x x = 5. x = + 5x x + = x 7. x 7x + = 0 given tat is a root 8. x + 7x x 8 = 0 given tat is a root..7 Solve te Absolute Value Inequality - write te solution as an interval. 9. x + 5 < 7 0. < 6 x ). x > 9. < x + 7. x +. x Polynomial Division 5. x 0x x + ) x + ) 6. x ) x ) 7. x + 0x + 8x 7x + ) x) 8. x + 6x + 6x 0x ) x ) 9. x + 6x + 6x 0x x + x 0. x + x x 5 x + x + 5. x 5 8x + x + x x +. x + 6x + x x +
4 ..9 Function Notation - evaluate te function as indicated, and simplify.. fx) = 7x a) f ) b) f ) c) ft) f ) d) fx + ). fx) = x x 5 a) f0) b) f 5 ) c) f) f ) d) fx + ) 5. gx) = x + x + 6. fx) = a) g0) b) g ) c) g) + g 5) d) fx ) { x + 8, x < 0 0 x, x 0 a) f) b) f 0) c) f0) d) f6) f ), x < 7. fx) = 5x, x < 0x, x a) f ) b) f) c) f) + f) d) f 5) + f ) 8. fx) = x + 5 a) fx + ) f) b) fx + ) fx) c) fx ) f) x d) fx + ) fx) In exercises 9-, find and simplify te difference quotient fx + ) fx) for eac function. 9. fx) = x + x fx) = x x +. fx) = x. fx) = x..0 Function Operations In exercises -6, below find a) f g)x) b) g f)x) c) f g)). fx) = x +, gx) = x. fx) = x, gx) = x + 5. fx) = 6x, gx) = x fx) = 5x, gx) = x + x In exercises 7-50, A) Find an equation for f, te inverse function. B) Verify tat your equation is correct by sowing tat ff x)) = x and f fx)) = x. 7. fx) = x 8. fx) = 8x + 9. fx) = x fx) = x x + Solve eac inequality in exercises 5-5, and express eac solution set in interval notation. 5. x x 0 5. x + x 5. x x + x x 0 5. x x x x 0 < 0 algebra review.dvi
5 . Answers to Review Exercises.. Answers for: Simplify eac expression x x 0. 75x 7. x y. y x). x y. y 9 5. y x 6. x y 5 7. x 8y 8. 9 x y.. Answers for: Factor te expression completely. 9. y y + ) 0. xyx + y). x + )x ). + y) y). 9a + b )9a b ). x + )x + )x ) 5. z )z + 6z + 9) 6. x + ) 7. 55x + )x + ) 8. u + 7)u ) 9. y 6)y + ) 0. x )x + ). x 5)x 9). x )x ). x 5y)x + y). 6x + 7y)6x 7y) 5. 8x ) 6. 5xx + )x ) 7. x + )x + )x ) 8. 5 x + ) x ) 9. x 5)x + 5)x + ) 0. y y + )y ). x + )x )y ). x )6x 5)x + ). x + ) 6x 7x + 6). x + )6x 5) x 0x + 6) 5. x + ) x ) x + 0x + 8) 6. 8x ) 9 x ) x 5x ) 7. 7x ) x + 5) x + ) / 8. x + 9 x + ) / 9. x )x + )x ) 50. x + y) x + y 0)x + y + 0) 5. x )x )x + ) 5. x + )x )x )
6 .. Answers for: Complete te Square 5. x + ) 9 5. ) t y ) x ) x ) x ).. Answers for: Perform te Indicated Operations 59. 5x + 0 x x x 0 6. x + 6 x 8 6. y + 5y 6. x + x + 9 xx 9) 6. x 0x + x + )x )x 5) 65. 8x + 0x 6 x x + ) 66. x xx + ) x + 5 x sr r s x x x + + )x + ), x +, xx + ) x 5x + ) x xx + ) 79. x 5)8x + ) x + ) 80. x + ) 5 9x 89) x 5) 8. x + 9) x + x 9) x + 6) 8. xx + ) x ) x x + ) x 5) 5 8. x + x + ) 85. xx ) x ) 6x x + ) x + )..5 Answers for: Rationalize te Numerator 87. x + y) x + y), x y 88. x + ) + + x +, 0
7 ..6 Answers for: Solve te Equation 89. x = or x = 90. x = or x = 9. No solutions - false 9. All R reals) 9. x = 9. x = 5 or x = 95. x =, x = 0, or x = x = 0, x = 7, or x = 97. x = ± x = ± No real solutions 00. x = ± x = or x = 0. x = or x = 0. z = or z = 0. x =, x =, x = 0, or x = 05. x = 06. x = x = 08. t = 09. No solution x ± ) 0. q = p S) S p). q = fp p f RR R. R = R R RR RR. x = x=-0 is extraneous). x = x=-6 is extraneous) 5. x = 9 6. x = 8 7. x = or x = ± 5 8. x =, x =, or x =..7 Answers for: Solve te Absolute Value Inequality 9., ) 0. 7, 9)., 8) or 6, )., ). [0, 8]., 5] or [, )..8 Answers for: Polynomial Division 5. x x x + 6. x + x + x 7. 5x x + x 7 x 8. x + 5 x x x ) 9. x + x + 0. x. x x + + x x +. x + x x + 0 x + x + 5
8 ..9 Answers for: Function Notation. a) 0 b) c) 0 7t d) 7x a) 0 b) c) 5 d) a) b) Undefined c) d) x + x x x a) b) c) 0 d) 8 7. a) b) 0 c) d) 8. a) x + 8x b) x + c) x x 0 d) x +, x + +, 0 0. x, 0. x x x + ), 0. x + x, 0..0 Answers for: Function Operations. a) fgx)) = 6x 8x + b) gfx)) = x + c) fg)) =. a) fgx)) = x + b) gfx)) = x + c) fg)) = 5. a) fgx)) = x b) gfx)) = x c) fg)) = 6. a) fgx)) = 5x + 0x 7 b) gfx)) = 5x + 0x c) fg)) = 8 7. f x) = x + 8. f x) = x 9. f x) = x f x) = x x, x 5. [ 5, ) 5., ], ) 5., ) S [, ] S, ) 5., ), 5) algebra review.dvi
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