Re: January 7, 015 Math 080: Final Exam Review Page 1 of 6 Note: If you have difficulty with any of these problems, get help, then go back to the appropriate sections and work more problems! 1. Solve for x, writing solutions of inequalities in interval notation. (a) 1 5 (x 5) = 1 (x + ) 7 10 x + 1 + 3 x 1 = x 1 (c) 1 3x < 7 (d) x 3 6 = 1 (e) a = 3 xt (f) 5 x < 7 (g) x + x = 15 (h) ln x = 3 (i) x x + = 0 (j) 6 < x + 1 1 3 (k) 8 x = 1 16 (l) 3 x + 1 = x (m) log 8 (x) + log 8 (x + 3) = 1 (n) x + 1 5 (o) 3x x = 0 (p) x+1 = 3 (q) 3x x + 9 5 (r) 5 x + 3 x + = 1 (s) x x 1 or 3 (t) e x =. Solve each system. { 5x y = (a) x + 3y = 13 x + y z = 3x 5y + z = 3 6x 7y z = (c) x + y + z = 57 3x + y = 3 x z = 6 3. Perform the indicated operations and/or simplify as much as possible. (a) (8x + 1) (8x 3 9x 6x) ( x + 11x 6) (c) (x + 1)(x x + 1) (d) (e) 1 + a b 1 a b ( 3x ) ( x 5 ) x ( 3x 6) (f) x 1 x x + 1 x + x 1 (g) 6 m m m + m (h) x + x 3 x 5 3 x (i) (j) (k) 3 x + x 1 x 3 + 9x + 0x x3 + x y x y a 1 + 3 a + (l) (m) ( 6x ) 13x 5 ( 1 5x + x ( ) x 5 5x + 1 ) (n) 3 3 50 (o) 8 0 500 (p) m 10 m 3 (q) ( 8) ( a) 3 (r) ( x 1 + y 1) ( ) 5x 5 1/ (s) 16x 3 (t) ( x 3 8x + 9x 1 ) (x 3) (u) (16) 3/ (7) 1/3 (v) x 10x + 5 (w) 3 80a 7 b 11 (x) 3a 8 b 3
Re: January 7, 015 Math 080: Final Exam Review Page of 6. Simplify by rationalizing the denominator. (a) 6 7 + (c) 3 a 3 3b x 6x (d) 5 + 7 5 7 5. Find the exact value of each logarithm. (a) log 3 81 log 100 (c) log ( 1 8 ) 6. Find, algebraically, the vertex of the parabola f(x) = x + x + 5. Find the maximum or minimum value of the function and indicate whether the value is the maximum or whether it is the minimum. 7. Find, algebraically, the x-intercept(s) of the graph of f(x) = x + 3x + 1. Give exact, simplified values and then round your answers to two decimal places. 8. Solve by completing the square: x + 9x 5 = 0. 9. Find the domain (in interval notation) of f(x) = 7 + x x. 10. Find the domain (in interval notation) of g(x) = x + 3. 11. Factor completely. (a) x + 3x 3 8x 5x 0xy + 16y (c) 1000x 3 + y 3 (d) x 7x (e) 6x 11y (f) 5c 3 16d 3 1. Graph on a rectangular coordinate system (use graph paper) and state the domain and range of each function by reading it from your graph. (a) 3x y = 1 f(x) = (x + ) + 9 (c) g(x) = 3 (d) p(x) = x 3 (e) F (x) = x 3x + (f) x = 13. Find the slope and y-intercept of 3x y =. 1. Find the slope of the line passing through the points ( 6, ) and (8, 3). 15. Write the equation of the line passing through the point (, 5) and (a) parallel to 3x + y = 5 perpendicular to x y = 1. 16. If f(x) = x 7 and g(x) = x + 3, find (f g)(x) and simplify. 17. If f(x) = 1 x + 3, find a formula for f 1 (x). 18. Find f(5) if f(x) = x x x 7 and simplify. 19. Find f(x + h) if f(x) = 3x + 1 and simplify. 0. Use properties of logarithms to write as a single logarithm: 5 log a 3 log b 1 log c.
Re: January 7, 015 Math 080: Final Exam Review Page 3 of 6 1. Use properties of logarithms to write as a sum/difference of logarithms. Write powers as factors: log ( 3x y ). Perform the indicated operations with the complex numbers. (a) (7 i)(5 + i) 1 3i + i 3. An airplane can travel 300 miles against the wind in the same time it travels 00 miles with the wind. If the speed of the wind is 5 mph, what is the speed of the plane in still air?. Using straight line depreciation, the value V of a particular piece of office equipment x years after purchase is given by the linear function V (x) = 5, 00 675x for 0 x 8. (a) Identify the independent and dependent variables and explain what each represents. What is the domain of this function? (c) What is the value of this office equipment 3 years after it is purchased? (d) After how long will the value of this equipment be depreciate completely, i.e., $0? 5. An object falls from a window 00 feet high. It s height above the ground is given by the function d(t) = 00 16t, where t is in seconds and d is in feet. (a) Find the height of the object after seconds. How long does it take the object to hit the ground? 6. A rectangular floor can be covered by 11 square feet of carpeting. Find the dimensions of the rectangular floor if its width is 1 foot less than half its length. 7. A 750 square foot apartment rents for $1550 and a 1050 square foot apartment rents for $195. Suppose that there is a linear relationship between the size (number of square feet) of an apartment and its rent. (a) Find a linear function that expresses the cost C of renting an apartment as a function of its size s. What is the slope and what does it mean in terms of the size of the apartment and its rent? (c) Predict the rent of a 900 square foot apartment. (d) If the rent of an apartment is $1800, what is its size (number of square feet)? 8. A student drives 0 miles at a constant speed, but she will be late for her Intermediate Algebra final exam unless she drives 0 mph faster for the last 30 miles. If her total travel time is 1 hour, what is her original speed? 9. Using data from 1960 to 000, the function N(x) = 71.0x+836.83 represents the approximate number of registered climbers at Mt. Rainier x years after 1960. (Source: National Parks Service, U.S. Dept of the Interior) (a) Identify the independent and dependent variables and explain what each represents. Evaluate N(3) and explain what it represents. (c) There were 971 registered climber on Mt. Rainier in 003. Compare this value to your result in part and comment on any differences. 30. A man rows 0 miles downstream and return in 13 hours and 0 minutes. If he can row mph in still water, what is the rate of the current? 31. A businessman received a total of $1,0 in interest for the year on two investments. If one of the investments paid 7% annual interest and the other paid 1% annual interest, how much was invested at each rate if the total amount of money invested was $1,800? 3. If each person works alone, it takes one clerk twice as long as another clerk to stuff a batch of envelopes. Together, the two clerks can do the job in 3 hours. How long does it take each clerk, alone, to stuff the envelopes? 33. A company offers two types of health plans to its employees. Plan A pays 90% of an employee s medical bills after a $00 deductible, while Plan B pays 80% of an employee s medical bills after a $50 deductible. For what amounts of medical bills is Plan A better for an employee than Plan B?
Re: January 7, 015 Math 080: Final Exam Review Page of 6 3. A new car has a sticker price of $3,590. Suppose that the dealer markup on this car is 15%. What was the dealer s cost, to the nearest dollar? 35. The outside dimensions of a picture frame are 0 inches by 3 inches. The area of the picture within the frame is 1008 square inches. Find the width of the frame. 36. An investor split $60,000 among three banks. He received 5%, 6%, and 7% in interest on the three deposits. In the account earning 7% interest, he deposited twice as much as in the account earning 5% interest. If his total earnings were $3760, how much did he deposit in each bank? 37. It takes hours for a boat to travel 8 miles downstream. The same boat can travel 18 miles upstream in 3 hours. Find the speed of the current and the speed of the boat in still water. 38. A small company found that when their product is sold for a price of x dollars, the weekly revenue in dollars as a function of the price x is R(x) = 0.5x + 170x. For what selling price will the weekly revenue be maximized, and what is the maximum weekly revenue? 39. Based on data from the Kelly Blue Book, the value V of a Dodge Stratus that is t years old can be modeled by the function V (t) = 19, 8(0.8) t. According to the model, when will the car be worth $1,000? Round your answer to the nearest hundredth. 0. A wood craftsman makes children s rocking horses. He sells each rocking horse for $95. His monthly fixed costs of operating his business are $500, and it costs $35 in materials for each rocking horse. (a) Find the revenue function R treating the number of rocking horses x as the independent variable. Find the cost function C treating the number of rocking horses x as the independent variable. (c) Find the break-even number of rocking horses that must be manufactured and sold. What is the revenue and cost of this number of rocking horses?
Re: January 7, 015 Math 080: Final Exam Review Page 5 of 6 Answers 1. (a) 63 no solution (c) (, ) (d), 1 (e) 3a t (f) (, 1) (g) 5, 3 (h) e 3 (i) 1 ± i ( (j) 19 ], 1 (k) 3 (l) (m) 1 ( (n), 3 ] [1, ) (o) ±1, ± i 6 3 (p) log 3 log 1 or log 3 1.585 [ ) (q) (, ] 3, (r) 3 ± 19 (s) (, 8] (t) ln 1.85. (a) (, 3) (3,, 1) (c) ( 60, 183, 66) 3. (a) 6x + 16x + 1 [ ) 7, 8x 3 5x 17x + 6 (c) x 3 + 1 b (d) b a (e) 3x (f) x + 1 x + m + 1 (g) m(m 1)(m + ) (h) x 3 x 3 (i) x(x 3) (x + 5)(x y) (j) 5a + 1 (a 1)(a + ) (k) 1 8 (l) 3x + 1 x (m) 16 (n) (o) i 5 1 (p) m 7 (q) 8 a 3 (r) x y x + xy + y (s) 5x (t) x x + 3 + 8 x 3 (u) (v) x 5 (w) a b 3 3 10ab (x) a b 3. (a) 7 6x 3 3 18ab (c) 3b (d) 16 + 5 7 9 5. (a) (c) 3 6. vertex = (1, 7); (1, 7) is a maximum 7. 3 + 17 8. 9 ± 101 0.8 and 3 17 1.78 9. {x x, } or (, ) (, ) (, ) 10. [ 3 ), 11. (a) (x + 3)(x )(x + x + ) (5x y)
Re: January 7, 015 Math 080: Final Exam Review Page 6 of 6 (c) (10x + y)(100x 10xy + y ) (c) D : {x all real numbers }, R : {y 3} (d) D : {x all real numbers }, R : {y y > 3} { (e) D : {x all real numbers }, R : y y 1 } (c) $1737.50 (d) (x + 1)(x ) (e) ( (d) 950 square feet 8x + 11y )( 8x 11y ) 8. 0 mph (f) (3c d)(9c + 6cd + d ) 9. (a) independent variable is x, which represents the 1. (a) D : {x all real numbers }, R : {y all real numbers } D : {x all real numbers }, R : {y y 9} number of years since 1960; dependent variable is N, which represents the number of registered climbers at Mt. Rainier 1, 507; this represents the number of registered climbers in the year 003 (c) leave to student (f) D : {x }, R : {y all real numbers } 13. m = 3, y-intercept: (0, 1) 1. 1 1 15. (a) y = 3 x + y = x + 1 16. (f g)(x) = x + 1x + 17. f 1 (x) = x 6 18. 10 19. 3x + 3h + 1 ( ) a 5 0. log b 3 c 1. log 3 + log x 1 log y 30. mph 31. $590 at 7%; $6880 at 1% 3..5 hours; 9 hours 33. $1600 3. $3,590 35. inches 36. $16,000 at 5%; $1,000 at 6%; $3,000 at 7% 37. current is mph; still water is 10 mph 38. price is $30; revenue is $8,900 39..7 years 0. (a) R(x) = 95x C(x) = 35x + 500 (c) 75 rocking horses; $715. (a) 51 + 8i 1 17 13 17 i 3. 175 mph.. (a) independent variable is x, which represents the time in number of years; dependent variable is V, which represents the value of the office equipment [0, 8] (c) $3375 (d) after 8 years 5. (a) 336 ft 5 seconds 6. The length is 16 ft and the width is 7 ft. 7. (a) C = 1.5s + 61.5 m = 1.5; As the apartment increases in size by one square foot, the cost of renting an apartment is increasing by $1.5.