Chpter Prerequisites for Clculus. Functions nd Grphs Wht ou will lern out... Functions Domins nd Rnges Viewing nd Interpreting Grphs Even Functions nd Odd Functions Smmetr Functions Defined in Pieces Asolute Vlue Function Composite Functions nd wh... Functions nd grphs form the sis for understnding mthemtics nd pplictions. Functions The vlues of one vrile often depend on the vlues for nother: The temperture t which wter oils depends on elevtion (the oiling point drops s ou go up). The mount which our svings will grow in er depends on the interest rte offered the nk. The re of circle depends on the circle s rdius. In ech of these emples, the vlue of one vrile quntit depends on the vlue of nother. For emple, the oiling temperture of wter,, depends on the elevtion, e; the mount of interest, I, depends on the interest rte, r. We cll nd I dependent vriles ecuse the re determined the vlues of the vriles e nd r on which the depend. The vriles e nd r re independent vriles. A rule tht ssigns to ech element in one set unique element in nother set is clled function. The sets m e sets of n kind nd do not hve to e the sme. A function is like mchine tht ssigns unique output to ever llowle input. The inputs mke up the domin of the function; the outputs mke up the rnge (Figure.8). Input (Domin) f f() Output (Rnge) Figure.8 A mchine digrm for function. DEFINITION Function A function from set D to set R is rule tht ssigns unique element in R to ech element in D. In this definition, D is the domin of the function nd R is set contining the rnge (Figure.9). Leonhrd Euler (77 783) Leonhrd Euler, the dominnt mthemticl figure of his centur nd the most prolific mthemticin ever, ws lso n stronomer, phsicist, otnist, nd chemist, nd n epert in orientl lnguges. His work ws the first to give the function concept the prominence tht it hs in mthemtics tod. Euler s collected ooks nd ppers fill 7 volumes. This does not count his enormous correspondence to pproimtel 3 ddresses. His introductor lger tet, written originll in Germn (Euler ws Swiss), is still ville in English trnsltion. D domin set R rnge set Figure.9 A function from set D to set R. Not function. The ssignment is not unique. Euler invented smolic w to s is function of : = f(), which we red s equls f of. This nottion enles us to give different functions different nmes chnging the letters we use. To s tht the oiling point of wter is function of elevtion, we cn write = f(e). To s tht the re of circle is function of the circle s rdius, we cn write A = A(r), giving the function the sme nme s the dependent vrile. D R
Section. Functions nd Grphs 3 The nottion = f() gives w to denote specific vlues of function. The vlue of f t cn e written s f( ), red f of. EXAMPLE The Circle-Are Function Write formul tht epresses the re of circle s function of its rdius. Use the formul to find the re of circle of rdius in. If the rdius of the circle is r, then the re A( r) of the circle cn e epressed s A(r) = pr. The re of circle of rdius cn e found evluting the function Ar ( ) t r =. A() = p() = 4p The re of circle of rdius is 4p in. Now Tr Eercise 3. Nme: The set of ll rel numers Nottion: or (, ) Nme: The set of numers greter thn Nottion: or (, ) Nme: The set of numers greter thn or equl to Nottion: or [, ) Nme: The set of numers less thn Nottion: or (, ) Nme: The set of numers less thn or equl to Nottion: or (, ] Figure. Infinite intervls rs on the numer line nd the numer line itself. The smol q (infinit) is used merel for convenience; it does not men there is numer q. Domins nd Rnges In Emple, the domin of the function is restricted contet: The independent vrile is rdius nd must e positive. When we define function = f() with formul nd the domin is not stted eplicitl or restricted contet, the domin is ssumed to e the lrgest set of -vlues for which the formul gives rel -vlues the so-clled nturl domin. If we wnt to restrict the domin, we must s so. The domin of = is understood to e the entire set of rel numers. We must write =, 7 if we wnt to restrict the function to positive vlues of. The domins nd rnges of mn rel-vlued functions of rel vrile re intervls or comintions of intervls. The intervls m e open, closed, or hlf-open (Figures. nd.) nd finite or infinite (Figure.). Nme: Open intervl Nottion: < < or (, ) Nme: Closed intervl Nottion: or [, ] Figure. Open nd closed finite intervls. Closed t nd open t Nottion: < or [, ) Open t nd closed t Nottion: < or (, ] Figure. Hlf-open finite intervls. The endpoints of n intervl mke up the intervl s oundr nd re clled oundr points. The remining points mke up the intervl s interior nd re clled interior points. Closed intervls contin their oundr points. Open intervls contin no oundr points. Ever point of n open intervl is n interior point of the intervl. Viewing nd Interpreting Grphs The points (, ) in the plne whose coordintes re the input-output pirs of function = f() mke up the function s grph. The grph of the function = +, for emple, is the set of points with coordintes (, ) for which equls +.
4 Chpter Prerequisites for Clculus EXAMPLE Identifing Domin nd Rnge of Function Identif the domin nd rnge, nd then sketch grph of the function. = = The formul gives rel -vlue for ever rel -vlue ecept =. (We cnnot divide n numer.) The domin is (- q, ) h (, q). The vlue tkes on ever rel numer ecept =. ( = c Z if = >c). The rnge is lso (- q, ) h(, q). A sketch is shown in Figure.3. 4 4 3 3 4 3 3 3 4 3 4 5 6 4 Figure.3 A sketch of the grph of = > nd =. (Emple ) The formul gives rel numer onl when is positive or zero. The domin is [, q). Becuse denotes the principl squre root of, is greter thn or equl to zero. The rnge is lso [, q). A sketch is shown in Figure.3. Now Tr Eercise 9. Grphing with pencil nd pper requires tht ou develop grph drwing skills. Grphing with grpher (grphing clcultor) requires tht ou develop grph viewing skills. Power Function An function tht cn e written in the form f () = k, where k nd re nonzero constnts, is power function. Grph Viewing Skills. Recognize tht the grph is resonle.. See ll the importnt chrcteristics of the grph. 3. Interpret those chrcteristics. 4. Recognize grpher filure. Being le to recognize tht grph is resonle comes with eperience. You need to know the sic functions, their grphs, nd how chnges in their equtions ffect the grphs. Grpher filure occurs when the grph produced grpher is less thn precise or even incorrect usull due to the limittions of the screen resolution of the grpher.
Section. Functions nd Grphs 5 EXAMPLE 3 Identifing Domin nd Rnge of Function Use grpher to identif the domin nd rnge, nd then drw grph of the function. = 4 - = >3 Figure.4 shows grph of the function for -4.7 4.7 nd -3. 3., tht is, the viewing window [-4.7, 4.7] [-3., 3.], with -scle = -scle =. The grph ppers to e the upper hlf of circle. The domin ppers to e [-, ]. This oservtion is correct ecuse we must hve 4 - Ú, or equivlentl, -. The rnge ppers to e [, ], which cn lso e verified lgericll. Grphing /3 Possile Grpher Filure = 4 = /3 On some grphing clcultors ou need to enter this function s = ( ) >3 or = ( >3 ) to otin correct grph. Tr grphing this function on our grpher. [ 4.7, 4.7] [ 3., 3.] [ 4.7, 4.7] [, 4] Figure.4 The grph of = 4 - nd = >3. (Emple 3) Figure.4 shows grph of the function in the viewing window [-4.7, 4.7] [-, 4], with -scle = -scle =. The domin ppers to e (- q, q), which we cn verif oserving tht >3 = (3 ). Also the rnge is [, q) the sme oservtion. Now Tr Eercise 5. (, ) O O (, ) 3 (, ) Even Functions nd Odd Functions Smmetr The grphs of even nd odd functions hve importnt smmetr properties. DEFINITIONS A function = f() is n Even Function, Odd Function for ever in the function s domin. even function of if f(-) = f(), odd function of if f(-) = -f(), (, ) Figure.5 The grph of = (n even function) is smmetric out the -is. The grph of = 3 (n odd function) is smmetric out the origin. The nmes even nd odd come from powers of. If is n even power of, s in = or = 4, it is n even function of ( ecuse (-) = nd (-) 4 = 4 ). If is n odd power of, s in = or = 3, it is n odd function of ( ecuse (-) = - nd (-) 3 = - 3 ). The grph of n even function is smmetric out the -is. Since f(-) = f(), point (, ) lies on the grph if nd onl if the point (-, ) lies on the grph (Figure.5). The grph of n odd function is smmetric out the origin. Since f(-) = -f(), point (, ) lies on the grph if nd onl if the point (-, -) lies on the grph (Figure.5).
6 Chpter Prerequisites for Clculus Equivlentl, grph is smmetric out the origin if rottion of 8 out the origin leves the grph unchnged. EXAMPLE 4 Recognizing Even nd Odd Functions f() = Even function: (-) = for ll ; smmetr out -is. f() = + Even function: (-) + = + for ll ; smmetr out -is (Figure.6). f() = f() = + Odd function: (-) = - for ll ; smmetr out the origin. Not odd: f(-) = - +, ut -f() = - -. The two re not equl. Not even: (-) + Z + for ll Z (Figure.6). Now Tr Eercises nd 3. It is useful in grphing to recognize even nd odd functions. Once we know the grph of either tpe of function on one side of the -is, we know its grph on oth sides. Functions Defined in Pieces While some functions re defined single formuls, others re defined ppling different formuls to different prts of their domins. Figure.6 When we dd the constnt term to the function =, the resulting function = + is still even nd its grph is still smmetric out the -is. When we dd the constnt term to the function =, the resulting function = + is no longer odd. The smmetr out the origin is lost. (Emple 4) EXAMPLE 5 Grphing Piecewise-Defined Functions -, 6 Grph = f() = u,, >. The vlues of f re given three seprte formuls: = - when 6, = when, nd = when 7. However, the function is just one function, whose domin is the entire set of rel numers (Figure.7). Now Tr Eercise 33. =, <,, > EXAMPLE 6 Writing Formuls for Piecewise Functions Write formul for the function = f() whose grph consists of the two line segments in Figure.8. [ 3, 3] [, 3] Figure.7 The grph of piecewisedefined function. (Emple 5) We find formuls for the segments from (, ) to (, ) nd from (, ) to (, ) nd piece them together in the mnner of Emple 5. Segment from (, ) to (, ) The line through (, ) nd (, ) hs slope m = ( - )>( - ) = nd -intercept =. Its slope-intercept eqution is =. The segment from (, ) to (, ) tht includes the point (, ) ut not the point (, ) is the grph of the function = restricted to the hlf-open intervl 6, nmel, =, 6. continued
Section. Functions nd Grphs 7 f () (, ) (, ) Segment from (, ) to (, ) The line through (, ) nd (, ) hs slope m = ( - )>( - ) = nd psses through the point (, ). The corresponding point-slope eqution for the line is = ( - ) +, or = -. The segment from (, ) to (, ) tht includes oth endpoints is the grph of = - restricted to the closed intervl, nmel, = -,. Figure.8 The segment on the left contins (, ) ut not (, ). The segment on the right contins oth of its endpoints. (Emple 6) Piecewise Formul Comining the formuls for the two pieces of the grph, we otin f() =, 6 -,. Now Tr Eercise 43. Asolute Vlue Function The solute vlue function = ƒ ƒ is defined piecewise the formul ƒ ƒ = -, 6, Ú. The function is even, nd its grph (Figure.9) is smmetric out the -is. 3 = 3 3 Figure.9 The solute vlue function hs domin (- q, q) nd rnge [, q). [ 4, 8] [ 3, 5] Figure. The lowest point of the grph of f() = ƒ - ƒ - is (, -). (Emple 7) EXAMPLE 7 Using Trnsformtions Drw the grph of f() = ƒ - ƒ -. Then find the domin nd rnge. The grph of f is the grph of the solute vlue function shifted units horizontll to the right nd unit verticll downwrd (Figure.). The domin of f is (- q, q) nd the rnge is [-, q). Now Tr Eercise 49. g g() f f(g()) Figure. Two functions cn e composed when portion of the rnge of the first lies in the domin of the second. Composite Functions Suppose tht some of the outputs of function g cn e used s inputs of function f. We cn then link g nd f to form new function whose inputs re inputs of g nd whose outputs re the numers f(g()), s in Figure.. We s tht the function f(g())
8 Chpter Prerequisites for Clculus (red f of g of ) is the composite of g nd f. It is mde composing g nd f in the order of first g, then f. The usul stnd-lone nottion for this composite is f which is red s f of g. Thus, the vlue of f t is (f g, g g)() = f(g()). EXAMPLE 8 Composing Functions Find formul for f(g()) if g() = nd f() = - 7. Then find f(g()). To find f(g()), we replce in the formul f() = - 7 the epression given for g(). f() = - 7 f(g()) = g() - 7 = - 7 We then find the vlue of f(g()) sustituting for. f(g()) = () - 7 = -3 Now Tr Eercise 5. EXPLORATION Composing Functions Some grphers llow function such s to e used s the independent vrile of nother function. With such grpher, we cn compose functions.. Enter the functions = f() = 4 -, = g() =, 3 = ( ()), nd 4 = ( ()). Which of 3 nd 4 corresponds to f g? to g f?. Grph,, nd 3 nd mke conjectures out the domin nd rnge of 3. 3. Grph,, nd 4 nd mke conjectures out the domin nd rnge of 4. 4. Confirm our conjectures lgericll finding formuls for 3 nd 4. Quick Review. (For help, go to Appendi A nd Section..) ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ Eercise numers with gr ckground indicte prolems tht the uthors hve designed to e solved without clcultor. In Eercises 6, solve for.. 3-5 + 3. ( - ) 7 3. - 3 4 4. - Ú 5 5. 6 6 6. 9 - Ú In Eercises 7 nd 8, descrie how the grph of f cn e trnsformed to the grph of g. 7. f() =, g() = ( + ) - 3 8. f() =, g() = - 5 + In Eercises 9, find ll rel solutions to the equtions. 9. f() = - 5 f() = 4 f() = -6. f() = > f() = -5 f() =. f() = + 7 f() = 4 f() =. f() = 3 - f() = - f() = 3
Section. Functions nd Grphs 9 Section. Eercises Eercise numers with gr ckground indicte prolems tht the uthors hve designed to e solved without clcultor. In Eercises 4, write formul for the function nd use the formul to find the indicted vlue of the function.. the re A of circle s function of its dimeter d; the re of circle of dimeter 4 in.. the height h of n equilterl tringle s function of its side length s; the height of n equilterl tringle of side length 3 m 3. the surfce re S of cue s function of the length of the cue s edge e; the surfce re of cue of edge length 5 ft 4. the volume V of sphere s function of the sphere s rdius r; the volume of sphere of rdius 3 cm In Eercises 5, identif the domin nd rnge nd sketch the grph of the function. 5. = 4-6. = - 9 7. = + - 8. = -- 9. =. = 4 - - 35. Writing to Lern The verticl line test to determine whether curve is the grph of function sttes: If ever verticl line in the -plne intersects given curve in t most one point, then the curve is the grph of function. Eplin wh this is true. 36. Writing to Lern For curve to e smmetric out the -is, the point (, ) must lie on the curve if nd onl if the point (, -) lies on the curve. Eplin wh curve tht is smmetric out the -is is not the grph of function, unless the function is =. In Eercises 37 4, use the verticl line test (see Eercise 35) to determine whether the curve is the grph of function. 37. 38.. = +. = + In Eercises 3, use grpher to identif the domin nd rnge nd drw the grph of the function. 39. 4. 3. = 3 4. = 3-5. = 3-6. = 9-7. = >5 8. = 3> 9. = 3-3. = 4 - In Eercises 3, determine whether the function is even, odd, or neither. Tr to nswer without writing nthing (ecept the nswer).. = 4. = + 3. = + 4. = - 3 5. = + 6. = + 3 7. = 8. = 3 - - 9. = 3. = - - In Eercises 3 34, grph the piecewise-defined functions. 3. f() = e 3 -, 3. f() = e, 6, 6, Ú 33. 3 4 -, 6 f() = c (3>) + 3>, 3 + 3, 7 3, 6 34. f() = c 3, -, 7 In Eercises 4 48, write piecewise formul for the function. 4. 4. (, ) 3 4 43. 44. 3 (, ) 5 (, ) 3
Chpter Prerequisites for Clculus 45. 46. (, ) (, ) 47. 48. 3 (T, ) A (, ) (, ) (3, ) (c) Use the qudrtic regression to predict the mount of revenue in. (d) Now find the liner regression for the dt nd use it to predict the mount of revenue in. 55. The Cone Prolem Begin with circulr piece of pper with 4-in. rdius s shown in. Cut out sector with n rc length of. Join the two edges of the remining portion to form cone with rdius r nd height h, s shown in. g() T f() (f g)()? - 5-5? + > (c) >? (d)? ƒ ƒ, Ú TABLE.5 Brodw Seson Revenue Yer T In Eercises 49 nd 5, drw the grph of the function. Then find its domin nd (c) rnge. 49. f() = - ƒ 3 - ƒ + 5. f() = ƒ + 4 ƒ - 3 In Eercises 5 nd 5, find f(g()) g(f()) (c) f(g()) (d) g(f()) (e) g(g(-)) (f) f(f()) 5. f() = + 5, g() = - 3 5. f() = +, g() = - 53. Cop nd complete the following tle. 54. Brodw Seson Sttistics Tle.5 shows the gross revenue for the Brodw seson in millions of dollrs for severl ers. Amount ($ millions) 994 46 999 63 4 769 5 86 6 939 7 938 Source: The Legue of Americn Thetres nd Producers, Inc., New York, NY, s reported in The World Almnc nd Book of Fcts, 9. A T T 3T T Eplin wh the circumference of the se of the cone is 8p -. Epress the rdius r s function of. (c) Epress the height h s function of. (d) Epress the volume V of the cone s function of. 56. Industril Costs Dton Power nd Light, Inc., hs power plnt on the Mimi River where the river is 8 ft wide. To l new cle from the plnt to loction in the cit mi downstrem on the opposite side costs $8 per foot cross the river nd $ per foot long the lnd. Suppose tht the cle goes from the plnt to point Q on the opposite side tht is ft from the point P directl opposite the plnt. Write function C() tht gives the cost of ling the cle in terms of the distnce. Generte tle of vlues to determine if the lest epensive loction for point Q is less thn ft or greter thn ft from point P. 8 ft 4 in. P Power plnt Q mi h (Not to scle) r 4 in. Dton Find the qudrtic regression for the dt in Tle.5. Let = 99 represent 99, = 99 represent 99, nd so forth. Superimpose the grph of the qudrtic regression eqution on sctter plot of the dt.
Section. Functions nd Grphs Stndrdized Test Questions 57. True or Flse The function f() = 4 + + is n even function. Justif our nswer. 58. True or Flse The function f() = -3 is n odd function. Justif our nswer. 59. Multiple Choice Which of the following gives the domin of f() = 9 -? (A) Z ;3 (B) (-3, 3) (C) [-3, 3] (D) (- q, -3) h (3, q) (E) (3, q) 6. Multiple Choice Which of the following gives the rnge of f() = + -? (A) (- q, ) h (, q) (B) Z (C) ll rel numers (D) (- q, ) h (, q) (E) Z 6. Multiple Choice If f() = - nd g() = + 3, which of the following gives (f g)()? (A) (B) 6 (C) 7 (D) 9 (E) 6. Multiple Choice The length L of rectngle is twice s long s its width W. Which of the following gives the re A of the rectngle s function of its width? (A) A(W) = 3W (B) A(W) = (C) A(W) = W W (D) A(W) = W + W (E) A(W) = W - W Eplortions In Eercises 63 66, grph f g nd g f nd mke conjecture out the domin nd rnge of ech function. Then confirm our conjectures finding formuls for f g nd g f. 63. f() = - 7, g() = 64. f() = -, g() = 65. f() = - 3, g() = + 66. f() = - 3 +, g() = + 3 - Group Activit In Eercises 67 7, portion of the grph of function defined on [-, ] is shown. Complete ech grph ssuming tht the grph is even, odd. 67. 68. [.5 69. 7. [ f ().3 Etending the Ides 7. Enter =, = - nd 3 = + on our grpher. Grph 3 in [-3, 3] [-, 3]. Compre the domin of the grph of 3 with the domins of the grphs of nd. (c) Replce 3 -, -, #, >, nd >, in turn, nd repet the comprison of prt. (d) Bsed on our oservtions in nd (c), wht would ou conjecture out the domins of sums, differences, products, nd quotients of functions? 7. Even nd Odd Functions Must the product of two even functions lws e even? Give resons for our nswer. Cn nthing e sid out the product of two odd functions? Give resons for our nswer..3.3