660_ch0pp00-075.qd 0/6/08 :8 PM Page. Functions and Their Representations Concept Eplanation Eamples Identifing graphs of functions Vertical Line Test: If ever vertical line intersects a graph at no more than one point, then the graph represents a function. (Otherwise the graph does not represent a function.) Not a function Verbal Words describe precisel what is computed. A verbal of ƒ() = is Square the input to obtain the output. Smbolic Numerical Graphical Mathematical formula Table of values Graph of ordered pairs (, ) that satisf = ƒ() The squaring function is given b ƒ() =, and the square root function is given b g() =. A partial numerical of ƒ() = is shown. 0 f() 0 6 9 Each point on the graph satisfies =. f() =. Eercises Evaluating and Representing Functions. If ƒ(-) =, identif a point on the graph of ƒ.. If ƒ() = -9.7, identif a point on the graph of ƒ.. If (7, 8) lies on the graph of ƒ, then ƒ ( ) =.. If (-, ) lies on the graph of ƒ, then ƒ ( ) =. Eercises 5 0: Graph = ƒ() b hand b first plotting points to determine the shape of the graph. 5. ƒ() = 6. ƒ() = - 7. ƒ() = 8. ƒ() = + 9. ƒ() = - 0. ƒ() = +. ƒ() = -. ƒ() = -
660_ch0pp00-075.qd 0/6/08 :8 PM Page CHAPTER Introduction to Functions and Graphs. ƒ() = ƒ - ƒ. ƒ() = ƒ 0.5 ƒ 5. 6. 5. ƒ() = ƒ ƒ 6. ƒ() = ƒ - ƒ 7. ƒ() = 8. ƒ() = 9. ƒ() = - 0. ƒ() = + Eercises : Complete the following. (a) Find ƒ() for the indicated values of, if possible. (b) Find the domain of ƒ.. ƒ() = for = -, 5. ƒ() = - for = 8, -. ƒ() = for = -, a +. ƒ() = - for = -, a + 5. ƒ() = 6 - for = -, a + 6. ƒ() = - 5 for = -, a + 5 7. ƒ() = -7 for = 6, a - 8. ƒ() = - + for =, - 9. ƒ() = for =, -7 0. ƒ() = - for =, a +. ƒ() = for =, a - 5-9. ƒ() = for = -, a + + g() = 7. 8. 9. 0. g() = g() = Eercises 6: Use the graph of the function ƒ to estimate its domain and range. Evaluate ƒ(0)... g() = g() = g() =. ƒ() = for =, a + -. ƒ() = for = 0, a - a + - Eercises 5 0: (Refer to Eample.) Use the graph to complete the following. (a) Find the domain of g. (b) Use the formula to evaluate g(-) and g(). (c) Use the graph of g to evaluate g(-) and g().
660_ch0pp00-075.qd 0/6/08 :8 PM Page. Functions and Their Representations.. 57. ƒ() = 5-58. ƒ() = ƒ ƒ 59. ƒ() = + 60. ƒ() = - Eercises 6 and 6: A function g is defined. (a) Write g as a set of ordered pairs. (b) Give the domain and range of g. 6. g(-) =, g(0) =, g() = -, g() = 6. g(-) = 5, g(0) = -5, g() = 5, g(8) = 0 5. 6. Eercises 6 and 6: Epress a function ƒ with the specified. 6. Cost of Driving In 008 the average cost of driving a new car was about 50 cents per mile. Give smbolic, graphical, and numerical s of the cost in dollars of driving miles. For the numerical use a table with =,,,, 5, 6. (Source: Associated Press.) Eercises 7 and 8: Diagrams Complete the following. (a) Evaluate ƒ(). (b) Write ƒ as a set of ordered pairs. (c) Find the domain and range of ƒ. 7. 8. f f 7 0 8 5 6. Counterfeit Mone It is estimated that nine out of ever one million bills are counterfeit. Give a numerical (table) of the predicted number of counterfeit bills in a sample of million bills where = 0,,, Á, 6. (Source: Department of the Treasur.) Identifing Functions Eercises 65 70: Does the graph represent a function? If so, determine the function s domain and range. 65. 66. Eercises 9 5: Graph = ƒ() in the viewing rectangle [-.7,.7, ] b [-.,., ]. (a) Use the graph to evaluate ƒ(). ( b) Evaluate ƒ() smbolicall. (c) Let = -, -, -, 0,,, and make a table of values for ƒ(). 9. ƒ() = 0.5 50. ƒ() = -.5 5. ƒ() = + 5. ƒ() = ƒ.6 - ƒ Eercises 5 60: Use ƒ() to determine verbal, graphical, and numerical s. For the numerical use a table with = -, -, 0,,. Evaluate ƒ(). 5. ƒ() = 5. ƒ() = - 5 67. 68. 55. ƒ() = ƒ + ƒ 56. ƒ() = 8
660_ch0pp00-075.qd 0/6/08 :8 PM Page CHAPTER Introduction to Functions and Graphs 69. 70. 87. + = 70 88. ( - ) + = 89. + = 90. = Eercises 9 96: Formulas Write a smbolic ( formula) for a function g that calculates the given quantit. Then evaluate g(0) and interpret the result. 9. The number of inches in feet 9. The number of quarts in gallons Eercises 7 7: Determine if the following operation describes a function. Eplain our answer. 7. Calculating the cube root of a number 7. Calculating our age 7. Listing the students who passed a given English eam 7. Finding the -values in the domain of a relation 75. Identification Numbers A relation takes a student s identification number at our college as input and outputs the student s name. Does this relation compute a function? Eplain. 76. Heights A relation takes a student s height rounded to the nearest inch as input and outputs the student s name with that height. Does this relation tpicall compute a function? Eplain. Eercises 77 8: Determine if S is a function. 77. S = {(, ), (, ), (, 5), (, )} 78. 79. 80. S = {(-, 7), (-, 7), (, 9), (6, 7), (0, 0)} S = {(a, ), (b, ), (c, ), (d, ), (e, )} S = {(a, ), (a, ), (b, 5), (-b, 7)} 8. S is given b the table. 0.5-0.5 8. S is given b the table. Eercises 8 90: Determine if is a function of. 8. = 8. = + 85. + = 86. = - 7 9. The number of dollars in quarters 9. The number of quarters in dollars 95. The number of seconds in das 96. The number of feet in miles Applications 97. Income and Education (Refer to Eample.) The function I computes median 00 individual annual earnings for females (in dollars) b educational attainment. This function is defined b I(N ) = 9,6, I(H ) = 6,09, I(B) =,68, and I(M ) = 5,6. (Source: Digest of Education Statistics, 005.) (a) Write I as a set of ordered pairs. (b) Give the domain and range of I. 98. Music and Digital Downloads Function P computes the percentage of total music sales that were digital downloads during a selected ear. This function is defined b P(00) = 0.5%, P(00) =.%, P(00) =.9%, P(005) = 5.7%, and P(006) = 9.%. (Source: Recording Industr Association of America.) (a) Write P as a set of ordered pairs. (b) Give the domain and range of P. 99. Going Green The average person uses 00 paper napkins in one ear. Write the formula for a function N that calculates the number of paper napkins that the average person uses in ears. Evaluate N() and interpret our result. 00. Going Green The average top-loading washing machine uses about 0 gallons of water per load of clothes. Write the formula for a function W that calculates the number of gallons of water used while washing loads of clothes. Evaluate W(0) and interpret our result.
660_ch0pp00-075.qd 0/6/08 :8 PM Page 5. Tpes of Functions 5 0. Air Temperature (Refer to Eample 6.) When the relative humidit is 00%, air cools 5.8 F for ever -mile increase in altitude. Give verbal, smbolic, graphical, and numerical s ƒ that computes this change in temperature for an increase in altitude of miles for 0. (Source: L. Battan.) 0. Crutch Length (Refer to Eample 7.) Determine the crutch length for someone 6 feet inches tall. For each -inch increase in a person s height, b how much does the recommended crutch length increase? 0. Distance to Lightning Find a formula for a function ƒ that computes the distance between an observer and a lightning bolt when the speed of sound is 50 feet per second. Evaluate ƒ(5) and interpret the result. 0. Distance to Lightning Give a reasonable domain for the function ƒ that ou found in Eercise 0. Graph ƒ over the domain that ou selected. What is the range of our function? (Note that answers ma var.) Writing about Mathematics 05. Eplain how ou could use a complete numerical (table) for a function to determine its domain and range. 06. Eplain in our own words what a function is. How is a function different from a relation?. Tpes of Functions Average wind speed (mph) Identif and use constant and linear functions Interpret slope as a rate of change Identif and use nonlinear functions Recognize linear and nonlinear data 0 9 8 7 6 5 0 Figure.6 Model 5 6 7 8 9 0 Month A Discrete Constant Introduction Functions are used to describe, or model, everthing from weather to new product specs, global warming, and U.S. population. New functions are created each da in the dnamic field of mathematics. Finding new functions whether to describe the wind speed in Hawaii or to calculate the memor requirements of an ipod requires creativit. This section discusses three common tpes of functions: constant, linear, and nonlinear. Constant Functions The monthl average wind speeds in miles per hour at Hilo, Hawaii, from Ma through December are listed in Table.. Table. Month Ma June Jul Aug Sept Oct Nov Dec Wind speed (mph) 7 7 7 7 7 7 7 7 Source: J. Williams, The Weather Almanac. A Discrete Function It is apparent that the monthl average wind speed is constant between Ma and December. These data can be described b a set ƒ of ordered pairs (, ), where is the month and is the wind speed. The months have been assigned the standard numbers. ƒ = {(5, 7), (6, 7), (7, 7), (8, 7), (9, 7), (0, 7), (, 7), (, 7)} The function ƒ is given b ƒ() = 7, where = 5, 6, 7, Á,. The output of ƒ never changes. We sa that ƒ is a constant function and models the data in Table.. The range of ƒ is R = {7}, and the domain of ƒ is D = {5, 6, 7, 8, 9, 0,, }. Since ƒ is defined onl at individual or discrete values of, ƒ is called a discrete function. The graph of a discrete function suggests a scatterplot, as shown in Figure.6.