Introduction to Functions. Definition of Function. Set A is the domain. Inputs: 1, 2, 3, 4, 5, 6 FIGURE

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1 3330_0104.qd 40 1/7/05 Capter :35 AM Page 40 Functions and Teir Graps Functions Wat ou sould learn Determine weter relations between two variables are functions. Use function notation and evaluate functions. Find te domains of functions. Use functions to model and solve real-life problems. Evaluate difference quotients. W ou sould learn it Functions can be used to model and solve real-life problems. For instance, in Eercise 100 on page 5, ou will use a function to model te force of water against te face of a dam. Introduction to Functions Man everda penomena involve two quantities tat are related to eac oter b some rule of correspondence. Te matematical term for suc a rule of correspondence is a relation. In matematics, relations are often represented b matematical equations and formulas. For instance, te simple interest I earned on $1000 for 1 ear is related to te annual interest rate r b te formula I 1000r. Te formula I 1000r represents a special kind of relation tat matces eac item from one set wit eactl one item from a different set. Suc a relation is called a function. Definition of Function A function f from a set A to a set B is a relation tat assigns to eac element in te set A eactl one element in te set B. Te set A is te domain (or set of inputs) of te function f, and te set B contains te range (or set of outputs). To elp understand tis definition, look at te function tat relates te time of da to te temperature in Figure Time of da (P.M.) 1 Temperature (in degrees C) FIGURE Set A is te domain. Inputs: 1,, 3, 4, 5, Lester Lefkowitz/Corbis Set B contains te range. Outputs: 9, 10, 1, 13, Tis function can be represented b te following ordered pairs, in wic te first coordinate (-value) is te input and te second coordinate (-value) is te output. 1, 9,, 13, 3, 15, 4, 15, 5, 1, 6, 10 Caracteristics of a Function from Set A to Set B 1. Eac element in A must be matced wit an element in B.. Some elements in B ma not be matced wit an element in A. 3. Two or more elements in A ma be matced wit te same element in B. 4. An element in A (te domain) cannot be matced wit two different elements in B.

2 3330_0104.qd 1/7/05 8:35 AM Page 41 Section 1.4 Functions 41 Have our students pa special attention to te concepts of function, domain, and range, because te will be used trougout tis tet and in calculus. Functions are commonl represented in four was. Four Was to Represent a Function 1. Verball b a sentence tat describes ow te input variable is related to te output variable. Numericall b a table or a list of ordered pairs tat matces input values wit output values 3. Grapicall b points on a grap in a coordinate plane in wic te input values are represented b te orizontal ais and te output values are represented b te vertical ais 4. Algebraicall b an equation in two variables To determine weter or not a relation is a function, ou must decide weter eac input value is matced wit eactl one output value. If an input value is matced wit two or more output values, te relation is not a function. Eample 1 Testing for Functions Determine weter te relation represents as a function of. a. Te input value is te number of representatives from a state, and te output value is te number of senators. b. c. Input, Output, a. Tis verbal description does describe as a function of. Regardless of te value of, te value of is alwas. Suc functions are called constant functions. b. Tis table does not describe as a function of. Te input value is matced wit two different -values. c. Te grap in Figure 1.48 does describe as a function of. Eac input value is matced wit eactl one output value. Now tr Eercise 5. Representing functions b sets of ordered pairs is common in discrete matematics. In algebra, owever, it is more common to represent functions b equations or formulas involving two variables. For instance, te equation is a function of. represents te variable as a function of te variable. In tis equation, is FIGURE 1.48

3 3330_0104.qd 1/7/05 8:35 AM Page 4 4 Capter 1 Functions and Teir Graps te independent variable and is te dependent variable. Te domain of te function is te set of all values taken on b te independent variable, and te range of te function is te set of all values taken on b te dependent variable. Bettmann/Corbis Historical Note Leonard Euler ( ), a Swiss matematician, is considered to ave been te most prolific and productive matematician in istor. One of is greatest influences on matematics was is use of smbols, or notation. Te function notation f was introduced b Euler. Understanding te concept of a function is essential. Be sure tat students understand function notation. Frequentl f is misinterpreted as f times rater tan f of. Eample Testing for Functions Represented Algebraicall Wic of te equations represent(s) as a function of? a. 1 b. 1 To determine weter is a function of, tr to solve for in terms of. a. Solving for ields 1 1. Write original equation. Solve for. To eac value of tere corresponds eactl one value of. So, is a function of. b. Solving for ields 1 1 ±. Write original equation. Add to eac side. Solve for. Te ± indicates tat to a given value of tere correspond two values of. So, is not a function of. Now tr Eercise 15. Function Notation Wen an equation is used to represent a function, it is convenient to name te function so tat it can be referenced easil. For eample, ou know tat te equation 1 describes as a function of. Suppose ou give tis function te name f. Ten ou can use te following function notation. Input Output Equation f f 1 Te smbol f is read as te value of f at or simpl f of. Te smbol f corresponds to te -value for a given. So, ou can write f. Keep in mind tat f is te name of te function, wereas f is te value of te function at. For instance, te function given b f 3 as function values denoted b f, f 0, f, and so on. To find tese values, substitute te specified input values into te given equation. For, f For 0, f For, f

4 3330_0104.qd 1/7/05 8:35 AM Page 43 Section 1.4 Functions 43 Altoug f is often used as a convenient function name and is often used as te independent variable, ou can use oter letters. For instance, f 4 7, f t t 4t 7, and g s s 4s 7 all define te same function. In fact, te role of te independent variable is tat of a placeolder. Consequentl, te function could be described b f 4 7. Eample 3 Evaluating a Function In Eample 3, note tat g is not equal to g g. In general, g u v g u g v. Students often ave difficult understanding ow to evaluate piecewisedefined functions. You ma want to use te following additional eamples to demonstrate evaluation of piecewisedefined functions. a. Evaluate at 0, 1, 3. f 1, 1 3, > 1 Because 0 is less tan or equal to 1, use f 1 to obtain f For 1, use f 1 to obtain f For 3, use f 3 to obtain f b. Evaluate at 0, 3, 5. f 3, < 7, 4 1, > 4 Because 0 is less tan, use f 3 to obtain f For 3, use f 7 to obtain f 3 7. For 5, use f 1 to obtain f Let g 4 1. Find eac function value. a. g b. g t c. g a. Replacing wit in g 4 1 ields te following. b. Replacing wit t ields te following. c. Replacing wit ields te following. Now tr Eercise 9. A function defined b two or more equations over a specified domain is called a piecewise-defined function. Eample 4 A Piecewise-Defined Function Evaluate te function wen, 0, and 1. f 1, 1, g g t t 4 t 1 t 4t 1 g Because is less tan 0, use f 1 to obtain f 1. For 0, use f 1 to obtain f For 1, use f 1 to obtain f < 0 0 Now tr Eercise 35.

5 3330_0104.qd 1/7/05 8:35 AM Page Capter 1 Functions and Teir Graps Tecnolog Use a graping utilit to grap te functions given b 4 and 4. Wat is te domain of eac function? Do te domains of tese two functions overlap? If so, for wat values do te domains overlap? Te Domain of a Function Te domain of a function can be described eplicitl or it can be implied b te epression used to define te function. Te implied domain is te set of all real numbers for wic te epression is defined. For instance, te function given b f 1 4 Domain ecludes -values tat result in division b zero. as an implied domain tat consists of all real oter tan ±. Tese two values are ecluded from te domain because division b zero is undefined. Anoter common tpe of implied domain is tat used to avoid even roots of negative numbers. For eample, te function given b f Domain ecludes -values tat result in even roots of negative numbers. is defined onl for 0. So, its implied domain is te interval 0,. In general, te domain of a function ecludes values tat would cause division b zero or tat would result in te even root of a negative number. Eample 5 Finding te Domain of a Function Find te domain of eac function. a. f : 3, 0,, 4, 0,,,, 4, b. c. Volume of a spere: V 4 3 r 3 d. 4 a. Te domain of f consists of all first coordinates in te set of ordered pairs. Domain 3,, 0,, 4 b. Ecluding -values tat ield zero in te denominator, te domain of g is te set of all real numbers ecept 5. c. Because tis function represents te volume of a spere, te values of te radius r must be positive. So, te domain is te set of all real numbers r suc tat r > 0. d. Tis function is defined onl for -values for wic 4 0. B solving tis inequalit (see Section.7), ou can conclude tat. So, te domain is te interval,. Now tr Eercise 59. In Eample 5(c), note tat te domain of a function ma be implied b te psical contet. For instance, from te equation V 4 3 r 3 g 1 5 ou would ave no reason to restrict r to positive values, but te psical contet implies tat a spere cannot ave a negative or zero radius.

6 3330_0104.qd 1/7/05 8:35 AM Page 45 r = 4 r Applications Eample 6 Te Dimensions of a Container Section 1.4 Functions 45 FIGURE 1.49 You work in te marketing department of a soft-drink compan and are eperimenting wit a new can for iced tea tat is sligtl narrower and taller tan a standard can. For our eperimental can, te ratio of te eigt to te radius is 4, as sown in Figure a. Write te volume of te can as a function of te radius r. b. Write te volume of te can as a function of te eigt. V r r r 4r 4 r 3 a. Write V as a function of r. b. V 3 Write V as a function of Now tr Eercise 87. Eample 7 Te Pat of a Baseball A baseball is it at a point 3 feet above ground at a velocit of 100 feet per second and an angle of 45º. Te pat of te baseball is given b te function f were and are measured in feet, as sown in Figure Will te baseball clear a 10-foot fence located 300 feet from ome plate? Heigt (in feet) f() FIGURE 1.50 Baseball Pat f() = Distance (in feet) Wen 300, te eigt of te baseball is f feet. So, te baseball will clear te fence. Now tr Eercise 93. In te equation in Eample 7, te eigt of te baseball is a function of te distance from ome plate.

7 3330_0104.qd 1/7/05 8:35 AM Page Capter 1 Functions and Teir Graps Number of veicles (in tousands) FIGURE 1.51 Number of Alternative-Fueled Veicles in te U.S. V Year (5 1995) t Eample 8 Alternative-Fueled Veicles Te number V (in tousands) of alternative-fueled veicles in te United States increased in a linear pattern from 1995 to 1999, as sown in Figure Ten, in 000, te number of veicles took a jump and, until 00, increased in a different linear pattern. Tese two patterns can be approimated b te function V t 18.08t t 10., 5 t 9 10 t 1 were t represents te ear, wit t 5 corresponding to Use tis function to approimate te number of alternative-fueled veicles for eac ear from 1995 to 00. (Source: Science Applications International Corporation; Energ Information Administration) From 1995 to 1999, use V t 18.08t From 000 to 00, use V t 38.0t Now tr Eercise 95. Difference Quotients One of te basic definitions in calculus emplos te ratio f f, 0. Tis ratio is called a difference quotient, as illustrated in Eample 9. Eample 9 Evaluating a Difference Quotient For f 4 7, find f f Now tr Eercise 79. f f. 4 4, 0 Te smbol in calculus. indicates an eample or eercise tat igligts algebraic tecniques specificall used

8 3330_0104.qd 1/7/05 :47 PM Page 47 Section 1.4 Functions 47 Activities 1. Evaluate f 3 for a. f 3 b. f 1 c. f f. Answers: a. 6 b. 4 c. 3. Determine weter is a function of Answer: No 3. Find te domain: f 3 1. Answer: All real numbers You ma find it easier to calculate te difference quotient in Eample 9 b first finding f, and ten substituting te resulting epression into te difference quotient, as follows. f f f Summar of Function Terminolog Function: A function is a relationsip between two variables suc tat to eac value of te independent variable tere corresponds eactl one value of te dependent variable. Function Notation: f f is te name of te function. is te dependent variable. is te independent variable. f is te value of te function at. Domain: Te domain of a function is te set of all values (inputs) of te independent variable for wic te function is defined. If is in te domain of f, f is said to be defined at. If is not in te domain of f, f is said to be undefined at. 4 4, 0 Range: Te range of a function is te set of all values (outputs) assumed b te dependent variable (tat is, te set of all function values). Implied Domain: If f is defined b an algebraic epression and te domain is not specified, te implied domain consists of all real numbers for wic te epression is defined. W RITING ABOUT MATHEMATICS Everda Functions In groups of two or tree, identif common real-life functions. Consider everda activities, events, and epenses, suc as long distance telepone calls and car insurance. Here are two eamples. a. Te statement, Your appiness is a function of te grade ou receive in tis course is not a correct matematical use of te word function. Te word appiness is ambiguous. b. Te statement, Your federal income ta is a function of our adjusted gross income is a correct matematical use of te word function. Once ou ave determined our adjusted gross income, our income ta can be determined. Describe our functions in words. Avoid using ambiguous words. Can ou find an eample of a piecewise-defined function?

9 3330_0104.qd 1/7/05 8:35 AM Page Capter 1 Functions and Teir Graps 1.4 Eercises VOCABULARY CHECK: Fill in te blanks. 1. A relation tat assigns to eac element from a set of inputs, or, eactl one element in a set of outputs, or, is called a.. Functions are commonl represented in four different was,,,, and. 3. For an equation tat represents as a function of, te set of all values taken on b te variable is te domain, and te set of all values taken on b te variable is te range. 4. Te function given b f 1, 4, < 0 0 is an eample of a function. 5. If te domain of te function f is not given, ten te set of values of te independent variable for wic te epression is defined is called te. f f 6. In calculus, one of te basic definitions is tat of a, given b, 0. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for tis section at In Eercises 1 4, is te relationsip a function? 1. Domain Range. Domain 3. Domain Range 4. In Eercises 5 8, does te table describe a function? Eplain our reasoning National League American League Cubs Pirates Dodgers Orioles Yankees Twins Domain (Year) Input value 0 1 Output value Range Range (Number of Nort Atlantic tropical storms and urricanes) Input value Output value Input value Output value Input value Output value In Eercises 9 and 10, wic sets of ordered pairs represent functions from A to B? Eplain. 9. A 0, 1,, 3 and B,, 0, 1, (a) 0, 1,,,, 0, 3, (b) 0,,,,,, 3, 0,, 1 (c) 0, 0,, 0,, 0, 3, 0 (d) 0,, 3, 0,, A a, b, c and B 0, 1,, 3 (a) a, 1, c,, c, 3, b, 3 (b) a, 1, b,, c, 3 (c), a, 0, a,, c, 3, b (d) c, 0, b, 0, a, 3

10 3330_0104.qd 1/7/05 8:35 AM Page 49 Section 1.4 Functions 49 Circulation of Newspapers In Eercises 11 and 1, use te grap, wic sows te circulation (in millions) of dail newspapers in te United States. (Source: Editor & Publiser Compan) Circulation (in millions) Year Morning Evening Is te circulation of morning newspapers a function of te ear? Is te circulation of evening newspapers a function of te ear? Eplain. 1. Let f represent te circulation of evening newspapers in ear. Find f 998. In Eercises 13 4, determine weter te equation represents as a function of In Eercises 5 38, evaluate te function at eac specified value of te independent variable and simplif. 5. f 3 (a) f (b) f 3 (c) f 1 6. g 7 3 (a) g 0 (b) g 7 3 (c) g s 7. V r 4 3 r 3 (a) V 3 (b) V 3 (c) V r 8. t t t (a) (b).5 (c) 9. f 3 (a) f 4 (b) f 0.5 (c) f f 8 (a) f 8 (b) f (c) f q 1 9 (a) q 0 (b) q 3 (c) 3. q t t 3 t (a) q (b) q 0 (c) 33. f (a) f (b) f (c) 34. f 4 (a) f (b) f (c) 35. f 1, < 0, 0 (a) f (b) f 0 (c) (a) f (b) f (c) (a) f 3 (b) f 4 (c) In Eercises 39 44, complete te table g f, 1, > 1 3 1, f 4,, (a) f (b) f 1 (c) 4 5, f 0, < < 1, > f 3 f g t 1 t 3 t t 4. f s s s s f s < 1 > q 3 q f 1 f f f f 3 f

11 3330_0104.qd 1/7/05 8:35 AM Page Capter 1 Functions and Teir Graps f 1 4, 0, > 0 f f 9, < 3 3, 3 f f 74. f 1 Eploration In Eercises 75 78, matc te data wit one of te following functions f c, g c, c, and r c and determine te value of te constant c tat will make te function fit te data in te table In Eercises 45 5, find all real values of suc tat f f f f 48. f f f f 3 5. f In Eercises 53 56, find te value(s) of f g. 53. f 1, g f 4, f 3 1, f 4, g g 1 g for wic In Eercises 57 70, find te domain of te function. 57. f g t s 3 t g f t t f f g s 1 6 f s 68. f s f f 5 9 In Eercises 71 74, assume tat te domain of f is te set A {,, 0, 1, }. Determine te set of ordered pairs tat represents te function f. 71. f 7. f In Eercises 79 86, find te difference quotient and simplif our answer g 1, f 1, f 5, f 3 3, f 4, f t 1 t, Undef f 5 f 5, 0 f f, 0 f f, 0 g g 3, 3 3 f f, 0 f t f, t 1 t f 5, f f 5, f 3 1, f f 8, Geometr Write te area A of a square as a function of its perimeter P. 88. Geometr Write te area A of a circle as a function of its circumference C. Te smbol in calculus. indicates an eample or eercise tat igligts algebraic tecniques specificall used

12 3330_0104.qd 1/7/05 8:35 AM Page 51 Section 1.4 Functions Maimum Volume An open bo of maimum volume is to be made from a square piece of material 4 centimeters on a side b cutting equal squares from te corners and turning up te sides (see figure). 91. Geometr A rigt triangle is formed in te first quadrant b te - and -aes and a line troug te point, 1 (see figure). Write te area A of te triangle as a function of, and determine te domain of te function (0, b) (, 1) ( a, 0) = 36 4 (, ) (a) Te table sows te volume V (in cubic centimeters) of te bo for various eigts (in centimeters). Use te table to estimate te maimum volume. Heigt, Volume, V (b) Plot te points, V from te table in part (a). Does te relation defined b te ordered pairs represent V as a function of? (c) If V is a function of, write te function and determine its domain. 90. Maimum Profit Te cost per unit in te production of a portable CD plaer is $60. Te manufacturer carges $90 per unit for orders of 100 or less. To encourage large orders, te manufacturer reduces te carge b $0.15 per CD plaer for eac unit ordered in ecess of 100 (for eample, tere would be a carge of $87 per CD plaer for an order size of 10). (a) Te table sows te profit P (in dollars) for various numbers of units ordered,. Use te table to estimate te maimum profit. Units, Profit, P Units, Profit, P (b) Plot te points, P from te table in part (a). Does te relation defined b te ordered pairs represent P as a function of? (c) If P is a function of, write te function and determine its domain. FIGURE FOR 91 FIGURE FOR 9 9. Geometr A rectangle is bounded b te -ais and te semicircle 36 (see figure). Write te area A of te rectangle as a function of, and determine te domain of te function. 93. Pat of a Ball Te eigt (in feet) of a baseball trown b a cild is were is te orizontal distance (in feet) from were te ball was trown. Will te ball fl over te ead of anoter cild 30 feet awa tring to catc te ball? (Assume tat te cild wo is tring to catc te ball olds a baseball glove at a eigt of 5 feet.) 94. Prescription Drugs Te amounts d (in billions of dollars) spent on prescription drugs in te United States from 1991 to 00 (see figure) can be approimated b te model d t 5.0t 37, 18.7t 64, were t represents te ear, wit t 1 corresponding to Use tis model to find te amount spent on prescription drugs in eac ear from 1991 to 00. (Source: U.S. Centers for Medicare & Medicaid Services) Amount spent (in billions of dollars) d 1 t 7 8 t Year (1 1991) t

13 3330_0104.qd 1/7/05 8:35 AM Page 5 5 Capter 1 Functions and Teir Graps 95. Average Price Te average prices p (in tousands of dollars) of a new mobile ome in te United States from 1990 to 00 (see figure) can be approimated b te model p t were t represents te ear, wit t 0 corresponding to Use tis model to find te average price of a mobile ome in eac ear from 1990 to 00. (Source: U.S. Census Bureau) Mobile ome price (in tousands of dollars) 0.18t 0.57t 7.3,.50t 1.3, p 0 t 7 8 t Year (0 1990) 96. Postal Regulations A rectangular package to be sent b te U.S. Postal Service can ave a maimum combined lengt and girt (perimeter of a cross section) of 108 inces (see figure). t (b) Write te revenue R as a function of te number of units sold. (c) Write te profit P as a function of te number of units sold. (Note: P R C) 98. Average Cost Te inventor of a new game believes tat te variable cost for producing te game is $0.95 per unit and te fied costs are $6000. Te inventor sells eac game for $1.69. Let be te number of games sold. (a) Te total cost for a business is te sum of te variable cost and te fied costs. Write te total cost C as a function of te number of games sold. (b) Write te average cost per unit C C as a function of. 99. Transportation For groups of 80 or more people, a carter bus compan determines te rate per person according to te formula Rate n 80, n 80 were te rate is given in dollars and n is te number of people. (a) Write te revenue R for te bus compan as a function of n. (b) Use te function in part (a) to complete te table. Wat can ou conclude? n R n Psics Te force F (in tons) of water against te face of a dam is estimated b te function F , were is te dept of te water (in feet). (a) Complete te table. Wat can ou conclude from te table? F (a) Write te volume V of te package as a function of. Wat is te domain of te function? (b) Use a graping utilit to grap our function. Be sure to use an appropriate window setting. (c) Wat dimensions will maimize te volume of te package? Eplain our answer. 97. Cost, Revenue, and Profit A compan produces a product for wic te variable cost is $1.30 per unit and te fied costs are $98,000. Te product sells for $ Let be te number of units produced and sold. (a) Te total cost for a business is te sum of te variable cost and te fied costs. Write te total cost C as a function of te number of units produced. (b) Use te table to approimate te dept at wic te force against te dam is 1,000,000 tons. (c) Find te dept at wic te force against te dam is 1,000,000 tons algebraicall Heigt of a Balloon A balloon carring a transmitter ascends verticall from a point 3000 feet from te receiving station. (a) Draw a diagram tat gives a visual representation of te problem. Let represent te eigt of te balloon and let d represent te distance between te balloon and te receiving station. (b) Write te eigt of te balloon as a function of d. Wat is te domain of te function?

14 3330_0104.qd 1/7/05 8:35 AM Page 53 Section 1.4 Functions Wildlife Te grap sows te numbers of treatened and endangered fis species in te world from 1996 troug 003. Let f t represent te number of treatened and endangered fis species in te ear t. (Source: U.S. Fis and Wildlife Service) f 003 f 996 (a) Find and interpret te result in te contet of te problem. (b) Find a linear model for te data algebraicall. Let N represent te number of treatened and endangered fis species and let 6 correspond to (c) Use te model found in part (b) to complete te table. N Number of treatened and endangered fis species Model It ft () Year (d) Compare our results from part (c) wit te actual data. (e) Use a graping utilit to find a linear model for te data. Let 6 correspond to How does te model ou found in part (b) compare wit te model given b te graping utilit? t Sntesis True or False? In Eercises 103 and 104, determine weter te statement is true or false. Justif our answer Te domain of te function given b f 4 1 is,, and te range of f is 0, Te set of ordered pairs 8,, 6, 0, 4, 0,,, 0, 4,, represents a function Writing In our own words, eplain te meanings of domain and range Tink About It Consider f and g 3. W are te domains of f and g different? In Eercises 107 and 108, determine weter te statements use te word function in was tat are matematicall correct. Eplain our reasoning (a) Te sales ta on a purcased item is a function of te selling price. (b) Your score on te net algebra eam is a function of te number of ours ou stud te nigt before te eam (a) Te amount in our savings account is a function of our salar. (b) Te speed at wic a free-falling baseball strikes te ground is a function of te eigt from wic it was dropped. Skills Review In Eercises , solve te equation t 3 t t 5 t In Eercises , find te equation of te line passing troug te pair of points. 113., 5, 4, , 0,, , 5, 3, , 3,, 3