1.5 Function Arithmetic

Transcription

1 76 Relations and Functions.5 Function Aritmetic In te previous section we used te newly deined unction notation to make sense o epressions suc as ) + 2 and 2) or a iven unction. It would seem natural, ten, tat unctions sould ave teir own aritmetic wic is consistent wit te aritmetic o real numbers. Te ollowin deinitions allow us to add, subtract, multiply and divide unctions usin te aritmetic we already know or real numbers. Function Aritmetic Suppose and are unctions and is in bot te domain o and te domain o. a Te sum o and, denoted +, is te unction deined by te ormula + )) ) + ) Te dierence o and, denoted, is te unction deined by te ormula )) ) ) Te product o and, denoted, is te unction deined by te ormula )) )) Te quotient o and, denoted, is te unction deined by te ormula ) ) ), provided ) 0. a Tus is an element o te intersection o te two domains. In oter words, to add two unctions, we add teir outputs; to subtract two unctions, we subtract teir outputs, and so on. Note tat wile te ormula + )) ) + ) looks suspiciously like some kind o distributive property, it is notin o te sort; te addition on te let and side o te equation is unction addition, and we are usin tis equation to deine te output o te new unction + as te sum o te real number outputs rom and. Eample.5.. Let ) and ) 3.. Find + ) ) 2. Find )2) 3. Find te domain o ten ind and simpliy a ormula or )).

2 .5 Function Aritmetic Find te domain o ten ind and simpliy a ormula or Solution. ) ).. To ind + ) ) we irst ind ) 8 and ) 4. By deinition, we ave tat + ) ) ) + ) To ind )2), we irst need 2) and 2). Since 2) 20 and 2) 5 2, our ormula yields )2) 2)2) 20) 5 2) One metod to ind te domain o is to ind te domain o and o separately, ten ind te intersection o tese two sets. Owin to te denominator in te epression ) 3, we et tat te domain o is, 0) 0, ). Since ) is valid or all real numbers, we ave no urter restrictions. Tus te domain o matces te domain o, namely,, 0) 0, ). A second metod is to analyze te ormula or )) beore simpliyin and look or te usual domain issues. In tis case, )) ) ) so we ind, as beore, te domain is, 0) 0, ). 3 ) ), Movin alon, we need to simpliy a ormula or )). In tis case, we et common denominators and attempt to reduce te resultin raction. Doin so, we et )) ) ) 3 ) ) et common denominators 4. As in te previous eample, we ave two ways to approac indin te domain o. First, we can ind te domain ) o and separately, and ind te intersection o tese two sets. In addition, since ) ) ), we are introducin a new denominator, namely ), so we need to uard aainst tis bein 0 as well. Our previous work tells us tat te domain o is, 0) 0, ) and te domain o is, ). Settin ) 0 ives

3 78 Relations and Functions or 0, 3. As a result, te domain o is all real numbers ecept 0 and 3, or, 0) 0, 3) 3, ). Alternatively, we may proceed as above and analyze te epression ) ) ) beore simpliyin. In tis case, ) ) ) We see immediately rom te little denominator tat 0. To keep te bi denominator away rom 0, we solve and et 0 or 3. Hence, as beore, we ind te domain o to be, 0) 0, 3) 3, ). Net, we ind and simpliy a ormula or ). ) ) ) ) 6 2 2) ) ) 3 ) ) 2 2 simpliy compound ractions actor cancel Please note te importance o indin te domain o a unction beore simpliyin its epression. In number 4 in Eample.5. above, ad we waited to ind te domain o until ater simpliyin, we d just ave te ormula to o by, and we would incorrectly!) state te domain as, 0) 0, ), 2 2 since te oter troublesome number, 3, was canceled away. We ll see wat tis means eometrically in Capter 4.

4 .5 Function Aritmetic 79 Net, we turn our attention to te dierence quotient o a unction. Deinition.8. Given a unction, te dierence quotient o is te epression + ) ) We will revisit tis concept in Section 2., but or now, we use it as a way to practice unction notation and unction aritmetic. For reasons wic will become clear in Calculus, simpliyin a dierence quotient means rewritin it in a orm were te in te deinition o te dierence quotient cancels rom te denominator. Once tat appens, we consider our work to be done. Eample.5.2. Find and simpliy te dierence quotients or te ollowin unctions. ) ) r) Solution.. To ind + ), we replace every occurrence o in te ormula ) 2 2 wit te quantity + ) to et + ) + ) 2 + ) So te dierence quotient is + ) ) ) 2 2 ) ) 2 + ) actor cancel 2 +.

5 80 Relations and Functions 2. To ind + ), we replace every occurrence o in te ormula ) 3 quantity + ) to et 2+ wit te wic yields + ) ) , + ) ) ) ) )2 + ) )2 + ) )2 + ) )2 + ) )2 + ) )2 + ) )2 + ) Since we ave manaed to cancel te oriinal rom te denominator, we are done. 3. For r), we et r + ) + so te dierence quotient is r + ) r) + In order to cancel te rom te denominator, we rationalize te numerator by multiplyin by its conjuate. 2 2 Rationalizin te numerator!? How s tat or a twist!

6 .5 Function Aritmetic 8 r + ) r) + ) ) ) Multiply by te conjuate ) 2 ) ) + ) + + ) + + ) + + ) + + Dierence o Squares. Since we ave removed te oriinal rom te denominator, we are done. As mentioned beore, we will revisit dierence quotients in Section 2. were we will eplain tem eometrically. For now, we want to move on to some classic applications o unction aritmetic rom Economics and or tat, we need to tink like an entrepreneur. 3 Suppose you are a manuacturer makin a certain product. 4 Let be te production level, tat is, te number o items produced in a iven time period. It is customary to let C) denote te unction wic calculates te total cost o producin te items. Te quantity C0), wic represents te cost o producin no items, is called te ied cost, and represents te amount o money required to bein production. Associated wit te total cost C) is cost per item, or averae cost, denoted C) and read C-bar o. To compute C), we take te total cost C) and divide by te number o items produced to et C) C) On te retail end, we ave te price p cared per item. To simpliy te dialo and computations in tis tet, we assume tat te number o items sold equals te number o items produced. From a 3 Not really, but entrepreneur is te buzzword o te day and we re tryin to be trendy. 4 Poorly desined resin Sasquatc statues, or eample. Feel ree to coose your own entrepreneurial antasy.

7 82 Relations and Functions retail perspective, it seems natural to tink o te number o items sold,, as a unction o te price cared, p. Ater all, te retailer can easily adjust te price to sell more product. In te lanuae o unctions, would be te dependent variable and p would be te independent variable or, usin unction notation, we ave a unction p). Wile we will adopt tis convention later in te tet, 5 we will old wit tradition at tis point and consider te price p as a unction o te number o items sold,. Tat is, we reard as te independent variable and p as te dependent variable and speak o te price-demand unction, p). Hence, p) returns te price cared per item wen items are produced and sold. Our net unction to consider is te revenue unction, R). Te unction R) computes te amount o money collected as a result o sellin items. Since p) is te price cared per item, we ave R) p). Finally, te proit unction, P ) calculates ow muc money is earned ater te costs are paid. Tat is, P ) R C)) R) C). We summarize all o tese unctions below. Summary o Common Economic Functions Suppose represents te quantity o items produced and sold. Te price-demand unction p) calculates te price per item. Te revenue unction R) calculates te total money collected by sellin items at a price p), R) p). Te cost unction C) calculates te cost to produce items. Te value C0) is called te ied cost or start-up cost. Te averae cost unction C) C) Here, we necessarily assume > 0. calculates te cost per item wen makin items. Te proit unction P ) calculates te money earned ater costs are paid wen items are produced and sold, P ) R C)) R) C). It is i time or an eample. Eample.5.3. Let represent te number o dopi media players dopis 6 ) produced and sold in a typical week. Suppose te cost, in dollars, to produce dopis is iven by C) , or 0, and te price, in dollars per dopi, is iven by p) or Find and interpret C0). 2. Find and interpret C0). 3. Find and interpret p0) and p20). 4. Solve p) 0 and interpret te result. 5. Find and simpliy epressions or te revenue unction R) and te proit unction P ). 6. Find and interpret R0) and P 0). 7. Solve P ) 0 and interpret te result. 5 See Eample in Section Pronounced dopeys...

8 .5 Function Aritmetic 83 Solution.. We substitute 0 into te ormula or C) and et C0) 000) Tis means to produce 0 dopis, it costs $2000. In oter words, te ied or start-up) costs are $2000. Te reader is encouraed to contemplate wat sorts o epenses tese mit be. 2. Since C) C) C0), C0) Tis means wen 0 dopis are produced, te cost to manuacture tem amounts to $300 per dopi. 3. Pluin 0 into te epression or p) ives p0) ) 450. Tis means no dopis are sold i te price is $450 per dopi. On te oter and, p20) ) 50 wic means to sell 20 dopis in a typical week, te price sould be set at $50 per dopi. 4. Settin p) 0 ives Solvin ives 30. Tis means in order to sell 30 dopis in a typical week, te price needs to be set to $0. Wat s more, tis means tat even i dopis were iven away or ree, te retailer would only be able to move 30 o tem To ind te revenue, we compute R) p) 450 5) Since te ormula or p) is valid only or 0 30, our ormula R) is also restricted to For te proit, P ) R C)) R) C). Usin te iven ormula or C) and te derived ormula or R), we et P ) ) ) As beore, te validity o tis ormula is or 0 30 only. 6. We ind R0) 0 wic means i no dopis are sold, we ave no revenue, wic makes sense. Turnin to proit, P 0) 2000 since P ) R) C) and P 0) R0) C0) Tis means tat i no dopis are sold, more money $2000 to be eact!) was put into producin te dopis tan was recouped in sales. In number, we ound te ied costs to be $2000, so it makes sense tat i we sell no dopis, we are out tose start-up costs. 7. Settin P ) 0 ives Factorin ives 5 0)3 40) 0 so 0 or Wat do tese values mean in te contet o te problem? Since P ) R) C), solvin P ) 0 is te same as solvin R) C). Tis means tat te solutions to P ) 0 are te production and sales) iures or wic te sales revenue eactly balances te total production costs. Tese are te so-called break even points. Te solution 0 means 0 dopis sould be produced and sold) durin te week to recoup te cost o production. For , tins are a bit more complicated. Even tou 3.3 satisies 0 30, and ence is in te domain o P, it doesn t make sense in te contet o tis problem to produce a ractional part o a dopi. 8 Evaluatin P 3) 5 and P 4) 40, we see tat producin and sellin 3 dopis per week makes a slit) proit, wereas producin just one more puts us back into te red. Wile breakin even is nice, we ultimately would like to ind wat production level and price) will result in te larest proit, and we ll do just tat... in Section Imaine tat! Givin sometin away or ree and ardly anyone takin advantae o it... 8 We ve seen tis sort o tin beore in Section.4..

9 84 Relations and Functions.5. Eercises In Eercises - 0, use te pair o unctions and to ind te ollowin values i tey eist. + )2) ) ) )) ) 2 0) 2). ) 3 + and ) 4 2. ) 2 and ) ) 2 and ) ) 2 3 and ) ) + 3 and ) 2 6. ) 4 and ) ) 2 and ) ) 2 and ) ) 2 and ) 2 0. ) 2 + and ) 2 + In Eercises - 20, use te pair o unctions and to ind te domain o te indicated unction ten ind and simpliy an epression or it. + )) )) )) ). ) 2 + and ) 2 2. ) 4 and ) 2 3. ) 2 and ) 3 4. ) 2 and ) 7 5. ) 2 4 and ) ) and ) ) 2 and ) 2 8. ) and ) 9. ) and ) ) 5 and ) ) 5 In Eercises 2-45, ind and simpliy te dierence quotient + ) ) 2. ) ) ) ) ) ) 4 2 or te iven unction.

10 .5 Function Aritmetic ) ) ) m + b were m ) a 2 + b + c were a 0 3. ) ) ) ) ) ) ) ) ) ) ) ) ) a + b, were a ) 45. ) 3. HINT: a b) a 2 + ab + b 2) a 3 b 3 In Eercises 46-50, C) denotes te cost to produce items and p) denotes te price-demand unction in te iven economic scenario. In eac Eercise, do te ollowin: Find and interpret C0). Find and interpret C0). Find and interpret p5) Find and simpliy R). Find and simpliy P ). Solve P ) 0 and interpret. 46. Te cost, in dollars, to produce I d rater be a Sasquatc T-Sirts is C) , 0 and te price-demand unction, in dollars per sirt, is p) 30 2, Te cost, in dollars, to produce bottles o 00% All-Natural Certiied Free-Trade Oranic Sasquatc Tonic is C) , 0 and te price-demand unction, in dollars per bottle, is p) 35, Te cost, in cents, to produce cups o Mountain Tunder Lemonade at Junior s Lemonade Stand is C) , 0 and te price-demand unction, in cents per cup, is p) 90 3, Te daily cost, in dollars, to produce Sasquatc Berry Pies C) , 0 and te price-demand unction, in dollars per pie, is p) 2 0.5, 0 24.

11 86 Relations and Functions 50. Te montly cost, in undreds o dollars, to produce custom built electric scooters is C) , 0 and te price-demand unction, in undreds o dollars per scooter, is p) 40 2, In Eercises 5-62, let be te unction deined by and let be te unction deined { 3, 4), 2, 2),, 0), 0, ),, 3), 2, 4), 3, )} { 3, 2), 2, 0),, 4), 0, 0),, 3), 2, ), 3, 2)}. Compute te indicated value i it eists ) 3) 52. )2) 53. ) ) )) 55. )3) 56. ) 3) 57. 2) 58. ) 59. 2) 60. ) ) 6. ) 3) 62. ) 3)