SECTION 2.4 Library of Functions; Piecewise-defined Functions 109 In Problems 43–46, the graph of a piecewise-defined function is given. Write a definition for each function. 43. x y 2 22 (21, 1) (0, 0) (2, 1) 2 44. x y 2 22 2 (0, 0) (2, 1) (21, 21) 45. x y 2 22 (2, 0) (1, 1) (21, 1) (0, 0) 46. x y 22 (2, 2) (1, 1) (21, 0) (0, 2) 2 47. If f x x int 2 , ( ) ( ) = find (a) f 1.7 ( ) (b) f 2.8 ( ) (c) f 3.6 ( ) − Applications and Extensions 49. (a) Graph ( ) ( ) = − ≤ < − + ≤ ≤ ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ f x x x x x 1 if 0 2 2 10 if 2 6 2 (b) Find the domain of f. (c) Find the absolute maximum and the absolute minimum, if they exist. 50. (a) Graph ( ) = − + − ≤ < = + < ≤ ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ f x x x x x x 1if 2 0 2 if 0 1 if 0 2 (b) Find the domain of f. (c) Find the absolute maximum and the absolute minimum, if they exist. 51. Rental Car Rates Maya’s car is in the shop for repairs that should take one day to complete. She found a local car rental company that charges a fee of $17 each day plus 13 cents per mile for the first 150 miles, and then charges 27 cents per mile for miles in excess of 150 miles. The following function is used to calculate the daily cost of driving the car x miles: ( ) = + ≤ ≤ − > ⎧ ⎨ ⎪⎪ ⎩⎪⎪ C x x x x x 0.13 17 if 0 150 0.4 23.5 if 150 Compute the cost of each of the following number of miles driven in one day: (a) 76 (b) 133 (c) 189 52. Parking at O’Hare International Airport The short-term (no more than 24 hours) parking fee F (in dollars) for parking x hours on a weekday at O’Hare International Airport’s main parking garage can be modeled by the function ( ) ( ) ( ) = − − < ≤ − − < ≤ < ≤ < ≤ ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ ⎪ F x x x x x x x 3 3 int 1 if 0 2 6 4 int 2 if 2 4 36 if 4 8 77 if 8 24 Determine the fee for parking in the short-term parking garage for (a) 0.5 hour (b) 2 hours (c) 3.25 hours (d) 8 hours Source: O’Hare International Airport. 53. Cupcake Catering A bakery uses the following pricing for large orders of cupcakes: The first 50 cupcakes of any order cost $2.00 each. The next 150 cupcakes cost only $1.50 each. Each cupcake ordered in excess of 200 costs $1.15 each. (a) If a mom orders 2 dozen cupcakes for her child’s birthday party, what will be the total cost for her order? (b) If a couple orders 120 cupcakes for their wedding reception, what will be the total cost of their order? (c) Develop a function that models the cost C for an order of x cupcakes. (d) Graph the function you found in part (c). 54. Alternative Tax Plan A proposed alternative to the current federal income tax system for single taxpayers, based on annual taxable income, is explained as follows: For the first $75,000 of income, every dollar is taxed at a rate of 8%. Each additional dollar over $75,000 is taxed at a rate of 16%, for the next $125,000 of income. Each additional dollar over $200,000 is taxed at a rate of 22%, for the next $150,000 of income. Every dollar over $350,000 is taxed at a rate of 30%. (a) If a person earns $48,000 in one year, how much federal income tax will they pay? (b) If a person earns $132,000 in one year, how much federal income tax will they pay? (c) If a person earns $231,000 in one year, how much federal income tax will they pay? (d) Develop a function T that specifies the yearly federal income tax for a person who earns x dollars per year. (e) Graph the function you found in part (d). 48. If f x x int 2 , ( ) ( ) = find (a) f 1.2 ( ) (b) f 1.6 ( ) (c) f 1.8 ( ) − 34. ( ) = + <− − − ≥− ⎧ ⎨ ⎪⎪ ⎩⎪⎪ f x x x x x 3 if 2 2 3 if 2 35. ( ) = + − ≤ < = − + > ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ f x x x x x x 3 if 2 1 5 if 1 2 if 1 36. ( ) = + − ≤ < − = − > ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ f x x x x x x 2 5 if 3 0 3 if 0 5 if 0 37. ( ) = + < ≥ ⎧ ⎨ ⎪⎪ ⎩⎪⎪ f x x x x x 1 if 0 if 0 2 38. ( ) = < ≥ ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ f x x x x x 1 if 0 if 0 3 39. ( ) = − ≤ < > ⎧ ⎨ ⎪⎪ ⎩⎪⎪ f x x x x x if 2 0 if 0 3 40. ( ) = − − ≤ < > ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ f x x x x x 2 if 3 1 if 1 41. ( ) = < ≤ + < < ≥ ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ f x x x x x x if 0 2 2 if 2 5 7 if 5 2 42. ( ) = + − ≤ < ≤ ≤ + > ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ f x x x x x x 3 5 if 3 0 5 if 0 2 1 if 2 2
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