Section. Functions and Function Notation 99 Section. Eercises. The amount of garbage, G, produced by a city with population p is given by G f( p). G is measured in tons per week, and p is measured in thousands of people. a. The town of Tola has a population of 40,000 and produces tons of garbage each week. Epress this information in terms of the function f. b. Eplain the meaning of the statement f 5. The number of cubic yards of dirt, D, needed to cover a garden with area a square feet is given by D g( a). a. A garden with area 5000 ft requires 50 cubic yards of dirt. Epress this information in terms of the function g. b. Eplain the meaning of the statement g 00. Let f () t be the number of ducks in a lake t years after 990. Eplain the meaning of each statement: a. f 5 0 b. f 0 40 4. Let ht () be the height above ground, in feet, of a rocket t seconds after launching. Eplain the meaning of each statement: a. 00 h 50 h b. 5. Select all of the following graphs which represent y as a function of. a b c d e f
400 Chapter 6. Select all of the following graphs which represent y as a function of. a b c d e f 7. Select all of the following tables which represent y as a function of. a. 5 0 5 b. 5 0 5 c. 5 0 0 y 8 4 y 8 8 y 8 4 8. Select all of the following tables which represent y as a function of. a. 6 b. 6 6 c. 6 y 0 0 y 0 4 y 0 4 9. Select all of the following tables which represent y as a function of. a. y b. y c. y d. y 0 - - -4 0-5 - -4 4 6 5 4 4 4 8 9 8 7 9 8 9 7 6 0. Select all of the following tables which represent y as a function of. a. y b. y c. y d. y -4 - -5 - - - - -5 6 4 4 5 4 5 9 7 7 9 9 8 8 7 6 0 4
Section. Functions and Function Notation 40. Select all of the following tables which represent y as a function of and are one-toone. a. 8 b. 8 c. 8 8 y 4 7 7 y 4 7 y 4 7. Select all of the following tables which represent y as a function of and are one-toone. a. 8 8 b. 8 4 c. 8 4 y 5 6 y 5 6 6 y 5 6. Select all of the following graphs which are one-to-one functions. a. b. c. d. e. f. 4. Select all of the following graphs which are one-to-one functions. a b c d e f
40 Chapter Given the each function f ( ) graphed, evaluate f () and f () 5. 6. 7. Given the function g ( ) graphed here, a. Evaluate g () b. Solve g 8. Given the function f ( ) graphed here. a. Evaluate f 4 b. Solve f( ) 4 9. Based on the table below, a. Evaluate f () b. Solve f ( ) 0 4 5 6 7 8 9 f ( ) 74 8 5 56 6 45 4 47 0. Based on the table below, a. Evaluate f (8) b. Solve f( ) 7 0 4 5 6 7 8 9 f ( ) 6 8 7 8 86 7 70 9 75 4 For each of the following functions, evaluate: f, f ( ), f (0), f (), and f (). f 4. f 8. f 8 7 4. f 6 7 4 4 5. f 6. f 5 7. f 8. f 4 9. f ( ) 0. f. f. f f. f 4.
Section. Functions and Function Notation 40 5. Suppose f 8 4. Compute the following: a. f ( ) f () b. f ( ) f () 6. Suppose f. Compute the following: a. f ( ) f (4) b. f ( ) f (4) 7. Let f t t 5 a. Evaluate f (0) b. Solve f t 0 8. Let gp6 p a. Evaluate g (0) b. Solve g p 0 9. Match each function name with its equation. a. y i. Cube root b. y ii. Reciprocal c. y iii. Linear iv. Square Root d. y v. Absolute Value vi. Quadratic e. y vii. Reciprocal Squared f. y viii. Cubic g. y h. y 40. Match each graph with its equation. a. y b. y c. y d. y e. y f. y g. y h. y i. ii. iii. iv. v. vi. vii. viii.
404 Chapter 4. Match each table with its equation. a. y i. In Out b. y - -0.5 - - c. y 0 _ d. y / e. y 0.5 0. f. y ii. In Out - - - - 0 0 iii. In Out - -8 - - 0 0 8 7 iv. In Out - 4-0 0 4 9 v. In Out - _ - _ 0 0 4 9 vi. In Out - - 0 0 4. Match each equation with its table a. Quadratic i. In Out b. Absolute Value - -0.5 c. Square Root d. Linear e. Cubic f. Reciprocal - 0 - _ 0.5 0. ii. In Out - - - - 0 0 iii. In Out - -8 - - 0 0 8 7 iv. In Out - 4-0 0 4 9 v. In Out - _ - _ 0 0 4 9 4. Write the equation of the circle centered at (, 9 ) with radius 6. 44. Write the equation of the circle centered at (9, 8 ) with radius. vi. In Out - - 0 0 45. Sketch a reasonable graph for each of the following functions. [UW] a. Height of a person depending on age. b. Height of the top of your head as you jump on a pogo stick for 5 seconds. c. The amount of postage you must put on a first class letter, depending on the weight of the letter.
Section. Functions and Function Notation 405 46. Sketch a reasonable graph for each of the following functions. [UW] a. Distance of your big toe from the ground as you ride your bike for 0 seconds. b. You height above the water level in a swimming pool after you dive off the high board. c. The percentage of dates and names you ll remember for a history test, depending on the time you study 47. Using the graph shown, a. Evaluate f ( c ) b. Solve f p c. Suppose f b z. Find f ( z ) d. What are the coordinates of points L and K? a b c K t r p L f() 48. Dave leaves his office in Padelford Hall on his way to teach in Gould Hall. Below are several different scenarios. In each case, sketch a plausible (reasonable) graph of the function s = d(t) which keeps track of Dave s distance s from Padelford Hall at time t. Take distance units to be feet and time units to be minutes. Assume Dave s path to Gould Hall is long a straight line which is 400 feet long. [UW] a. Dave leaves Padelford Hall and walks at a constant spend until he reaches Gould Hall 0 minutes later. b. Dave leaves Padelford Hall and walks at a constant speed. It takes him 6 minutes to reach the half-way point. Then he gets confused and stops for minute. He then continues on to Gould Hall at the same constant speed he had when he originally left Padelford Hall. c. Dave leaves Padelford Hall and walks at a constant speed. It takes him 6 minutes to reach the half-way point. Then he gets confused and stops for minute to figure out where he is. Dave then continues on to Gould Hall at twice the constant speed he had when he originally left Padelford Hall.
406 Chapter d. Dave leaves Padelford Hall and walks at a constant speed. It takes him 6 minutes to reach the half-way point. Then he gets confused and stops for minute to figure out where he is. Dave is totally lost, so he simply heads back to his office, walking the same constant speed he had when he originally left Padelford Hall. e. Dave leaves Padelford heading for Gould Hall at the same instant Angela leaves Gould Hall heading for Padelford Hall. Both walk at a constant speed, but Angela walks twice as fast as Dave. Indicate a plot of distance from Padelford vs. time for the both Angela and Dave. f. Suppose you want to sketch the graph of a new function s = g(t) that keeps track of Dave s distance s from Gould Hall at time t. How would your graphs change in (a)-(e)?
Section. Domain and Range 407 Section. Eercises Write the domain and range of the function using interval notation... Write the domain and range of each graph as an inequality.. 4. Suppose that you are holding your toy submarine under the water. You release it and it begins to ascend. The graph models the depth of the submarine as a function of time. What is the domain and range of the function in the graph? 5. 6.
408 Chapter Find the domain of each function 7. f 8. f 5 9. f 6 0. 5 0 f 9 6. f. f 6 8. f 4 4. f 5 4 5. f 4 4 6. f 5 6 f 7. 9 8. f 8 8 9 Given each function, evaluate: f ( ), f (0), f (), f (4) 7 if 0 4 9 if 0 9. f 0. f 76 if 0 48 if 0. f. if 4 5 if 5 if 0 f if 0 if. f 4. 4 if if if 0 f 4 if 0 if
Section. Domain and Range 409 Write a formula for the piecewise function graphed below. 5. 6. 7. 8. 9. 0. Sketch a graph of each piecewise function. f if 5 if. f 4 if 0 if 0. f 5. if 0 if 0 if f if if f 4. 6. if if if f if 0 if
40 Chapter Section. Eercises. The table below gives the annual sales (in millions of dollars) of a product. What was the average rate of change of annual sales a) Between 00 and 00 b) Between 00 and 004 year 998 999 000 00 00 00 004 005 006 sales 0 9 4 49 5 49 4. The table below gives the population of a town, in thousands. What was the average rate of change of population a) Between 00 and 004 b) Between 00 and 006 year 000 00 00 00 004 005 006 007 008 population 87 84 8 80 77 76 75 78 8. Based on the graph shown, estimate the average rate of change from = to = 4. 4. Based on the graph shown, estimate the average rate of change from = to = 5. Find the average rate of change of each function on the interval specified. 5. f ( ) on [, 5] 6. q( ) on [-4, ] 7. g ( ) on [-, ] 8. 9. k( t) 6t on [-, ] 0. t 4 h( ) 5 on [-, 4] t 4 p ( t) on [-, ] t Find the average rate of change of each function on the interval specified. Your answers will be epressions.. f ( ) 4 7 on [, b]. g ( ) 9 on [4, b]. h ( ) 4 on [, +h] 4. k ( ) 4 on [, +h] 5. 7. a ( t) on [9, 9+h] 6. b ( ) on [, +h] t 4 j( ) on [, +h] 8. r( t) 4t on [, +h] 9. f ( ) on [, +h] 0. g ( ) on [, +h]
Section. Rates of Change and Behavior of Graphs 4 For each function graphed, estimate the intervals on which the function is increasing and decreasing.... 4. For each table below, select whether the table represents a function that is increasing or decreasing, and whether the function is concave up or concave down. 5. f() 6. g() 7. h() 8. k() 90 00 0 4 70 90 5 8 80 70 5 4 6 4 75 4 40 4 5 5 7 5 00 5 5 9. f() -0-5 -7 4-47 5-54 0. g() -00-90 -60 4-00 5 0. h() -00-50 -5 4-0 5 0. k() -50-00 -00 4-400 5-900
4 Chapter For each function graphed, estimate the intervals on which the function is concave up and concave down, and the location of any inflection points.. 4. 5. 6. Use a graph to estimate the local etrema and inflection points of each function, and to estimate the intervals on which the function is increasing, decreasing, concave up, and concave down. 4 5 4 7. f ( ) 4 5 8. h ( ) 5 0 0 / 9. g ( t) t t 40. k( t) t t 4 4 4. m ( ) 0 4 4. n ( ) 8 8 6
Section.4 Composition of Functions 4 Section.4 Eercises Given each pair of equations, calculate f g 0 and g f 0. f 4 8, g7. f 5 7, g 4. f 4, g 4. f, g 4 Use the table of values to evaluate each epression 5. f( g (8)) 6. f g 5 7. g( f (5)) 8. g f 9. f( f (4)) 0. f f. gg ( ()) g g 6. f ( ) g( ) 0 7 9 6 5 5 6 8 4 4 5 0 8 6 7 7 8 9 4 9 0 Use the graphs to evaluate the epressions below.. f( g ()) 4. f g 5. g( f ()) 6. g f 0 7. f( f (5)) 8. f f 4 9. gg ( ()) g g 0 0. For each pair of functions, find 6 7 6 f g and g f. Simplify your answers. 4. f, g. f, g 4. f, g 4. f, g
44 Chapter 5. f, g5 6. f, g 4 7. If f 6, g ( ) 6 and h ( ), find f ( gh ( ( ))) 8. If f, g and h, find f ( gh ( ( ))) functions using interval notation. p a. Domain of m b. Domain of p( m ( )) c. Domain of mp ( ( )) 9. Given functions p and m functions using interval notation. q a. Domain of h b. Domain of qh ( ( )) c. Domain of hq ( ( )) 0. Given functions q and h 4, state the domains of the following 9, state the domains of the following. The function D( p ) gives the number of items that will be demanded when the price is p. The production cost, C ( ) is the cost of producing items. To determine the cost of production when the price is $6, you would do which of the following: a. Evaluate DC ( (6)) b. Evaluate CD ( (6)) c. Solve DC ( ( )) 6 d. Solve CDp ( ( )) 6. The function Ad ( ) gives the pain level on a scale of 0-0 eperienced by a patient with d milligrams of a pain reduction drug in their system. The milligrams of drug in the patient s system after t minutes is modeled by mt ( ). To determine when the patient will be at a pain level of 4, you would need to: a. Evaluate Am 4 b. Evaluate m A 4 c. Solve Amt d. Solve m Ad 4 4
Section.4 Composition of Functions 45. The radius r, in inches, of a balloon is related to the volume, V, by V rv ( ). Air 4 is pumped into the balloon, so the volume after t seconds is given by V t 0 0t a. Find the composite function rvt b. Find the time when the radius reaches 0 inches. 4. The number of bacteria in a refrigerated food product is given by NTT 56T, T where T is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by Tt ( ) 5t.5, where t is the time in hours. a. Find the composite function NTt b. Find the time when the bacteria count reaches 675 Find functions f ( ) and g ( ) so the given function can be epressed as h f g 5. h 6. h 5 4 7. h 8. h 5 9. h 40. h 4 4. Let f ( ) be a linear function, having form [UW] f f is a linear function a. Show that b. Find a function g() such that 4. Let f [UW] a. Sketch the graphs of g g 6 8 f a b for constants a and b. f, f f, f f f on the interval 0. b. Your graphs should all intersect at the point (6, 6). The value = 6 is called a fied point of the function f()since f (6) 6 ; that is, 6 is fied - it doesn t move when f is applied to it. Give an eplanation for why 6 is a fied point for any function f( f( f(... f( )...))). c. Linear functions (with the eception of f ( ) ) can have at most one fied point. Quadratic functions can have at most two. Find the fied points of the function g. d. Give a quadratic function whose fied points are = and =.
46 Chapter 4. A car leaves Seattle heading east. The speed of the car in mph after m minutes is 70m given by the function Cm. [UW] 0 m a. Find a function m f() s that converts seconds s into minutes m. Write out the formula for the new function C( f( s )); what does this function calculate? b. Find a function m g( h) that converts hours h into minutes m. Write out the formula for the new function Cgh ( ( )) ; what does this function calculate? c. Find a function z v( s) that converts mph s into ft/sec z. Write out the formula for the new function vcm ( ( ) ; what does this function calculate?
Section.5 Transformation of Functions 47 Section.5 Eercises Describe how each function is a transformation of the original function f ( ). f 49. f( 4). f( ) 4. f( 4) 5. f 5 6. f 8 7. f 8. f 7 9. f 0.. Write a formula for f ( ). Write a formula for f ( ) f 4 shifted up unit and left units shifted down units and right unit. Write a formula for f( ) shifted down 4 units and right units 4. Write a formula for f( ) shifted up units and left 4 units 5. Tables of values for f ( ), g, ( ) and h ( ) are given below. Write g ( ) and h ( ) as transformations of f ( ). - - 0 f() - - - - 0 g() - - - - - 0 h() - 0-6. Tables of values for f ( ), g, ( ) and h ( ) are given below. Write g ( ) and h ( ) as transformations of f ( ). - - 0 f() - - 4 - - - 0 g() - - 4 - - 0 h() - -4 0 The graph of f is shown. Sketch a graph of each transformation of f ( ) 7. g 8. h 9. w 0. q
48 Chapter Sketch a graph of each function as a transformation of a toolkit function. f t ( t). h 4. k 4. mt t Write an equation for the function graphed below 5. 6. 7. 8. Find a formula for each of the transformations of the square root whose graphs are given below. 9. 0.
Section.5 Transformation of Functions 49 The graph of f is shown. Sketch a graph of each transformation of f ( ). g. h. Starting with the graph of f 6 write the equation of the graph that results from a. reflecting f ( ) about the -ais and the y-ais b. reflecting f ( ) about the -ais, shifting left units, and down units 4. Starting with the graph of f 4 write the equation of the graph that results from a. reflecting f ( ) about the -ais b. reflecting f ( ) about the y-ais, shifting right 4 units, and up units Write an equation for the function graphed below 5. 6. 7. 8.
40 Chapter 9. For each equation below, determine if the function is Odd, Even, or Neither f a. 4 b. g ( ) c. h 40. For each equation below, determine if the function is Odd, Even, or Neither a. f b. 4 g h c. Describe how each function is a transformation of the original function f ( ) 4. f ( ) 4. f ( ) 4. 4 f ( ) 44. 6 f ( ) 45. f (5 ) 46. f ( ) 47. f 48. f 5 f 50. f ( ) 49. 5. Write a formula for f ( ) reflected over the y ais and horizontally compressed by a factor of 4 5. Write a formula for f ( ) reflected over the ais and horizontally stretched by a factor of 5. Write a formula for f( ) vertically compressed by a factor of, then shifted to the left units and down units. 54. Write a formula for f( ) vertically stretched by a factor of 8, then shifted to the right 4 units and up units. 55. Write a formula for f ( ) horizontally compressed by a factor of, then shifted to the right 5 units and up unit. 56. Write a formula for f ( ) horizontally stretched by a factor of, then shifted to the left 4 units and down units.
Section.5 Transformation of Functions 4 Describe how each formula is a transformation of a toolkit function. Then sketch a graph of the transformation. 57. f 4 5 58. g ( ) 5 59. h 4 60. k 6. m 6. n 6. p 64. q 4 a b 65. 4 66. 6 Determine the interval(s) on which the function is increasing and decreasing f g ( ) 5 67. 4 5 68. a k 69. 4 70. Determine the interval(s) on which the function is concave up and concave down 7. ( ) m 7. b 6 7. p k 74.
4 Chapter The function f ( ) is graphed here. Write an equation for each graph below as a transformation of f ( ). 75. 76. 77. 78. 79. 80. 8. 8. 8. 84. 85. 86.
Section.5 Transformation of Functions 4 Write an equation for the transformed toolkit function graphed below. 87. 88. 89. 90. 9. 9. 9. 94. 95. 96. 97. 98.
44 Chapter 99. Suppose you have a function y f( ) such that the domain of f ( ) is 6 and the range of f ( ) is y 5. [UW] a. What is the domain of f(( ))? b. What is the range of f ( ( ))? c. What is the domain of f( )? d. What is the range of f( )? e. Can you find constants B and C so that the domain of f ( B ( C)) is 8 9? f. Can you find constants A and D so that the range of Af( ) D is 0 y?
Section.6 Inverse Functions 45 Section.6 Eercises Assume that the function f is a one-to-one function.. If f (6) 7, find f (7). If f (), find f () f 4 8, find ( 8) f, find f ( ). If 5. If f 5, find f 5 7. Using the graph of f ( ) shown a. Find f 0 b. Solve f( ) 0 c. Find f 0 d. Solve f f 4. If 0 f, find f 6. If 4 8. Using the graph shown a. Find g () b. Solve g ( ) c. Find g () d. Solve g 9. Use the table below to fill in the missing values. 0 4 5 6 7 8 9 f() 8 0 7 4 6 5 9 a. Find f b. Solve f( ) c. Find f 0 d. Solve f 7
46 Chapter 0. Use the table below to fill in the missing values. t 0 4 5 6 7 8 h(t) 6 0 7 5 4 9 a. Find h 6 b. Solve ht () 0 c. Find h 5 d. Solve h t For each table below, create a table for. 6 9 4 f() 4 7 6 f.. 5 7 5 f() 6 9 6 For each function below, find f ( ). f 4. f 5 5. f 6. f 7. f 7 8. 9 0 f For each function, find a domain on which f is one-to-one and non-decreasing, then find the inverse of f restricted to that domain. 9. f 7 0. f 6 f 5. f.. If f 5 and g ( ) 5, find a. f ( g ( )) b. g( f( )) c. What does this tell us about the relationship between f ( ) and g? ( ) 4. If f( ) and a. f ( g ( )) b. g( f( )) g ( ), find c. What does this tell us about the relationship between f ( ) and g? ( )