Solve. 1) x 1 8 ) ( x ) x x 9 ) x 1 x 4) x + x 0 ) x + 9y 6) t t 4 7) y 8 4 x COLLEGE ALGEBRA FINAL REVIEW x 8) 81 x + 9) 4 7.07 x 10) 10 + 1e 10 11) solve for L P R K M + K L T 1) a) log x log( x+ 6) 1 b) log x+ log( x+ 4) log1+ log 1) log(4x 4) + 8 0 Write as a single logarithm and simplify, if possible. 14) log 0. 01 log1000 + 1) ln x + ln y ( lnx + lny) 16) Write as the sum of difference of logarithms; express powers as factors. Simplify the following: 17) ( x + ) ( x x + 1) ) ln x e ln xy z 18) ( y 4) 19) ( u + )( u ) ( u + 4) x y z 0) x y z ab 8xy 1) xy ab ) ( 1 1) 1 y x 4) 71 log t t 4x ) ln e 6) ( + i ) ( i)( + i) 7) + i i
Evaluate to decimal places. 8) log 7 9) e log e 0) 4 + ln 7 Given: log a. and log a 7. 807. Find the following to decimal places. 1) log a ) log a 7 ) loga 1 loga 4) log a 7 ) Convert to exponential equation ln x 6) Find the equation of the line (in slope-intercept form) that passes through (,-1) and (,-). 7) Find the distance between the points (7,-) and (8,-) Let f ( x) x x + 4. Find the following: 8) f ( ) 9) f ( t 1 ) 40) f ( x + h ) f ( x ) h Solve the following inequalities. Write the solutions in interval notation: 41) x 6x + 7 x 4) 1 4) x + 1 > xx ( + ) x + 1 x 46) > 1 4) 7 9 x 44) k > k + 47) For the function f( x) x 4x+ a) Find the vertex b) Find the line of symmetry c) Give the x, y coordinates of the maximum or minimum point d) Give the range of the function
Given the following cost and revenue functions, find the profit function and find the maximum profit as well as the number of units needed to obtain the maximum profit. 48) R ( x) x x ; C( x) x 8x 7 49) Sales of Packard Bell computers have grown exponentially. The total revenue function, in millions, is given by Rt ( ) 0.18(1.4) t where t is the number of years since 1990. Find Packard Bell s revenues in 1999, 00, and 00 assuming the growth rate remains the same. When will its sales revenue to $10 million? 0) A company finds that its costs follow the function Cx ( ).00x 1x+ 4,000 where x is the number of items made. How many items should it make to minimize its costs? 1) The sum of the width and length of a parallelogram is 90 cm. Find the dimensions for which the area is a maximum. Find all asymptotes of the functions. ) ) x + 4x + g ( x) x + 9 f ( x) x x x 11 4) Use your calculator to graph f( x) 1/( x 4) a) What is the y-intercept, if any? b) What is the x-intercept, if any? c) What is (are) the vertical asymptote(s), if any? d) What is the horizontal asymptote, if any? ) Find the center and the radius of the circle x + y 8x+ y+ 1 0. 6) What is the domain of f( x) x 7 7) What is the domain of f( x) x + x 16
8) Find the equation of the line (in slope-intercept form) that passes through (,-4) and is parallel to x y + 7 0. 9) Find the equation of the line passing through the point (-1,) and perpendicular to the line x y 9 x Given f ( x) and g( x) x 1. Find the following: 60) f ( x) + g( x) 6) ( f g)( x) 61) ( f g)( x) 1 6) f ( x ) 64) Given f( x) x and gx ( ) x 1. Find the following: a) ( f g)() b) ( fg)( 1) 1 c) ( f / g )( ) d) ( f g)( ) 6) Find the inverse of 6 f( x) x + 66) Find the inverse of f( x) x+ 4 Given f ( x) x x 1x + x + 10 67) Is - a zero of f ( x)? 68) Find all the zeros of f ( x) 69) Use synthetic division to determine whether is a zero of f( x) x + x x+ 6. 70) Find all the zeros of 4 f( x) x x x + 1x+ 10 71) Find all zeros of f( x) x x + 4x 0. Graph the following. State the domain, range, relative extrema, and intervals of increasing/decreasing. 7) f ( x) 8x x 7x +
7) Write the equation of the function which has the same shape as f( x) x but has been shifted 7 units to the right and up units and then reflected about the x axis. 74) Use your calculator to graph f( x ) 4 x. What is the domain? What is the range? 7) Suppose $,000 is invested at k percent, compounded continuously, and grows to $,98.6 in 7 years. a) What is k? b) How much will be in the account after 9 years? c) How long does it take the money to double? 76) A piece of charcoal is found to contain 0% of its carbon-14. If the decay rate of carbon-14 is -0.0001, how old is the piece of charcoal? 77) The volume V of a given mass of a gas varies directly as the temperature T and inversely as the pressure P. If V1 cubic centimeters when T 4º and P0 kg/square centimeters, what is the volume when T0º and P1 kg/square centimeters? Use the given table to answer questions 78 80 for a given rational function f( x) where f( x ) Y 1. 78) a) List the x-intercepts, if any, of f( x ). Write as an ordered pair. b) List any y-intercepts, if any, of f( x ). Write as an ordered pair. 79) Find the following: a) f ( 1) b) x such that f( x ) 4 80) Are there any restrictions for f( x )? If so, what are they? Explain.
1. 4.. 7 4. x, x 4. x ± 9 y 6. ± 7i 4 7. y 10 /, y COLLEGE ALGEBRA REVIEW ANSWERS 0. 1.. 7x 8y. x 1 6 1 z 1 8 18x yb 1 7a xy y x 4. 71. 4x 8. x 9. x log81 + log 4. 10. x 1.610 11. 1. a) log 7 log 4 019877. RKT L MKT + PT + K x b) x 1 1. x 01 14. log10 1 1. ln x y 1 16. 17. ln x+ ln y ln z 4x 7x+ 18. 9 y 4 y + 16 6. 6 7. i 8. 0.87 9. 1.611 0. 4.4 1. 4.644. 0. 48. 19. 4. 1.404. e x 6. y x 7 7. 10 8. 1 9. t 7t + 9 40. 4x + h 41. (, ] 4. (, ) (1, ) 4. [ 8,0] 19. u 4 16
44., and ( 1, ) 4. (,0) ( 0,1 ] 46., 11 47. a) (,6) b) x c) max(,6) d) y 6 48. units; $1 max. profit 49. 1999 $1.16 Million; 00 $4.817 Million; 00 $99.691 Million; 10 Million sometime in 1998 0. 100items 1. 4 cm. by 4 cm.. V.A.: x 9 ; H.A.: none S.A.: y x 1. V.A.: x, x ; H.A.: y 1 x 4. a) b)none c) x 4 d) as 4 y 0. C(4, 1), r 6. 7 x 7. x ± 4 8. y x 7 9. y x 1 60. 11 x 61. 10 x 9 x + 6. 10 x 7 1 x + 6. f ( x) 64. a) b) c) undefined d) -1 6. 66. f 67. Yes 1 6 ( x) [8, 0] x 1 f ( x) x 68.,,, 69. not a zero 70. 1,, 71., i, i 7. D: (, ) ; R: (, ) ; rel. max. of 6. @ x 0. 70; rel. min. of 0.97 @ x 0. 787 ; inc.: (, 0. 70) ( 0. 787, ) ( 0. 70, 0. 787) 7. x 7, ; dec.: 74. Domain: (, ) ; range: (0, ) 7. a) k.0714 b)$44.80 c) 1.1 yr 76. 10,0 years old 77. 0 cm 78. a) There are no x-intercepts in the table (where y 0). If this was a polynomial function, then the Intermediate Value Theorem says there s a zero between 1 and (but this is a rational function); b) (0,1.)
79. a) 1.6; b) 80. At x 4, there is an error for y. This means that 4 is not in the domain of f( x )