Chapter 1 Review Business Calculus 1

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1 Chpter Review Business Clculus Chpter : Review Section : Alger Review The following is list of some lger skills you re epected to hve. The emple prolems here re only rief review. This is not enough to tech you these skills. If you need more review, you cn look in ny ook clled Intermedite Alger. Lws of Eponents The Lws of Eponents let you rewrite lgeric epressions tht involve eponents. The lst three listed here re relly definitions rther thn rules. Lws of Eponents: All vriles here represent rel numers nd ll vriles in denomintors re nonzero. y y y y 0, provided 0 n n, provided 0, / n n provided this is rel numer Emple: Simplify s much s possile nd write your nswer using only positive eponents: 3 y Solution: y y y y Emple: Rewrite using only positive eponents: 3 5 / p

2 Chpter Review Business Clculus Solution: / 3 / / / 6 p p p 5 / 6 Writing Equtions of Lines p Identify the independent nd dependent vriles. The independent vrile is the one tht you think eplins the reltionship, nd the dependent vrile is the one tht you think responds. If you re counting cricket chirps per minute t vrious tempertures, the temperture could ffect how the crickets chirp, ut the cricket chirps re unlikely to ffect the temperture. In this emple, the independent vrile will e temperture nd the dependent vrile will e the numer of chirps per minute. Slope is numer tht tells you which direction the line points. If the slope is positive, the line points uphill s you red from left to right. If the slope is negtive, the line points downhill. Horizontl lines hve slope of zero. The closer the slope is to zero, the closer the line is to horizontl. The further the slope is from zero, either positive or negtive, the steeper the line is. Verticl lines hve undefined slope ecuse the run in the rise over run clcultion is zero. One wy to define stright line is s curve with constnt slope. You cn clculte slope using ny two points on the line. Prllel lines hve the sme slope. Perpendiculr lines hve negtive reciprocl slopes (tht is, their slopes multiply to mke ). Slope is rte of chnge. The units of slope re frctionl, y-units over -units, like miles per hour or dollrs per dy. In n ppliction prolem, look for the frctionl units to help you find the slope. The identifying feture of liner eqution is tht the slope is constnt. Equtions of lines: There re severl different forms of n eqution of line tht you might encounter. (Here I m ssuming is the independent vrile nd y is the dependent vrile.) Slope-Intercept form, y = m + : This is the fvorite of most students. The form is esy to rememer. You cn red the slope m nd y-intercept right off the eqution. If you don t hve the y-intercept, you will hve to do some lger to use this form. Point-Slope form, y y m : This is my fvorite form. The slope m is visile, nd is some known point on the line. I like this form the est ecuse there is no lger required just plop the slope nd one point into plce nd you re done. Stndrd form, A + By = C: This form is useful for compring different types of equtions. But it s not very helpful form for grphing or writing the eqution of line. You hve to do lger to find either the slope or ny point on the line., y

3 Chpter Review Business Clculus 3 All you need in order to write the eqution of line is the slope nd one point. The slope might e given to you (look for frctionl units!), or you might compute it from two points, or perhps get it from nother line tht is prllel or perpendiculr to it. The one point is usully given to you, or you could need to find the intersection of some curves to get the point. Find nd interpret the rte of chnge (slope). If you hve two points, whether they re given to you numericlly or if you red them off grph, you cn compute the slope using tht fmilir rise-over-run formul (the difference in the y s over the difference in the s). If you hve n eqution, you cn lgericlly mneuver it into slope-intercept form nd red the slope right off. If the sitution is descried in English, then the constnt rte of chnge is the slope. Rememer the units (y-units over -units)! Simply writing down sentence like the rte of chnge is 5 dollrs per yer is the iggest step in interpreting the slope. If the slope is positive, then the function is incresing. If the slope is negtive, then the function is decresing. If the slope is zero, the function is neither incresing or decresing (stying constnt). You cn compre the rtes of chnge for two functions y compring their slopes. The function whose grph is steeper, whose slope is further from zero, is chnging more rpidly. Find nd interpret vrious points of the liner function (for emple, the y-intercept). The points tht re on the line, the points tht stisfy the lgeric eqution, re individul emples tht fit your sitution. The - nd y- intercepts re usully importnt, ut they re not the only importnt points. The y-intercept is the plce where the line crosses the y-is. This is the y-vlue when = 0. The -intercept is the plce where the line crosses the -is. This is the -vlue tht mkes y = 0. There my e other importnt points tht rise ecuse of the pplied setting for your prolem. The units cn help you decide wht the importnt points men in ech prticulr sitution. Emple: The cost of diet progrm is $99 to join, plus $70 per week for the food. Descrie this function nd tell how much it would cost to join for ten weeks. Solution: You cn tell this is liner function, ecuse the rte of chnge (look for those frctionl units dollrs per week) is constnt. The slope of this function is 70 dollrs week this mens tht ech week I

4 Chpter Review Business Clculus 4 will spend nother 70 dollrs on the food. The independent vrile () is weeks nd the dependent vrile (y) is dollrs. The y-intercept is the $99 fee to join I will py this no mtter how mny weeks I elong. The -intercept for the line doesn t mke sense in this sitution. The line itself hs n -intercept, with some negtive numer of weeks. But it s not possile to join the diet progrm for negtive numer of weeks nd py zero dollrs. To sty with the progrm for ten weeks I would py $99 + 0($70) = $999. This is the point (, y) = (0, 999), which lies on the line. If you know the function is liner, two points re enough to write the formul. Use the two points to find the slope, nd then you cn solve to find the y-intercept. Emple: A fucet is dripping wter t constnt rte into owl. At :00, there ws ½ cup of wter in the owl. At :45, there ws ¾ cup of wter in the owl. How much wter will e in the owl t 3:30? Solution: This is liner function, ecuse the fucet is dripping t constnt rte. The domin is the set of times (hours pst noon). The rnge is the set of volumes in cups (numers 0). Let t e the time, mesured in hours pst noon, nd let W e the mount of wter in the owl, mesured in cups. There re two points given: when t =, W = 0.5, nd when t =.75, W = The slope is rise/run, ΔW/Δt = ( ) / (.75 ) =.5 /.75 = /3 cups per hour. So the eqution will e W t. 3 To find the W-intercept, just plug in one of the points you know nd solve for :, or 3 The function tht tells us how much wter is in the owl fter t hours is given y W t. 3 6 As check, let s mke sure this gives us the right nswer t the other known point if I plug in t =.75, I get W = 0.75, which is right. At 3:30, t = 3.5, nd W = 4/3 cup.. 6

5 Chpter Review Business Clculus 5 Fctoring nd the Qudrtic Formul By this time, you should hve seen how to solve qudrtic equtions in severl different wys. In this clss, you will only need couple: you will e epected to fctor esy things, use the qudrtic formul, nd to pproimte the solutions using technology (such s your clcultor). Fctor out common monomil fctors Emple: Recognize nd e le to fctor sum or difference of squres nd perfect squre trinomils. These specil products come up ll the time, nd you should e le to hndle them utomticlly. Fctor qudrtic trinomils with leding coefficient of if they re esy! This is guess-nd-check process; look for two numers whose sum is the coefficient of the liner term (eponent = ) nd whose product is the constnt term. Emple: Fctor 30 Solution: I m looking for two numers who dd to nd multiply to 30; 5 nd 6 work. So the fctoriztion is Unfortuntely, fctoring this wy cn tke long time, nd it s hrd to know if you should stop. If you cn t find fctoriztion, is it ecuse you didn t try the right fctors yet, or mye the fctors involve squre roots, or mye the qudrtic is lredy fully fctored? I usully don t spend very long serching for fctoriztion of qudrtic trinomil. If I cn t see fctoriztion quickly, I turn to the qudrtic formul (see elow). Qudrtic formul: The solutions of the qudrtic eqution c 0 (where,, nd c re rel numers nd 0 re given y 4c. Understnd nd frequently use tht deep connection etween roots nd fctors. This is often clled the Zero Product Property. This is the primry reson we fctor to find the roots, solutions, of n eqution. But rememer

6 Chpter Review Business Clculus 6 tht it goes the other wy, lso. If you know the solutions of polynomil eqution, you cn use them to construct the fctors. Use the qudrtic formul to give you the roots, nd use them to construct the fctors. If you cn t esily fctor qudrtic, you cn lwys eploit tht deep connection etween roots nd fctors. This tkes little it of time, too, ut it will lwys give you n nswer. This is lso the esiest wy to find fctors tht involve squre roots. Emple: Fctor Solution: I don t immeditely see fctoriztion, ut I cn use the qudrtic formul: 4c The two roots re nd , so the fctors re nd the leding coefficient is right: I ll need to multiply y 0 so tht Solving Eponentil Equtions You will need to rememer logrithms, ut you won t hve to do lot of lger with them. You won t hve to simplify epressions involving logrithms, so you won t need mny of the lws of logrithms. Here they re, just in cse you wnt to look t them the only ones you re likely to need re Lw 3 nd Lw 4. Lws of Eponents: : log y log log y : log log log y. In English: The log of product is the sum of the logs.. In English: The log of quotient is the difference of the logs. y n log nlog. In English: When you tke the log of power, the eponent comes down in 3: front. 4: log () = nd log () = 0 5: Chnge of Bse Formul: log log log

7 Chpter Review Business Clculus 7 An eponentil eqution is ny eqution tht involves n eponentil function. This is the technique you cn use to solve for the eponent.. Do s much ordinry lger (dding, sutrcting, multiplying or dividing lwys to oth sides of the eqution) s you cn in order to isolte the eponent.. Tke logrithm of oth sides. You cn use ny se you wnt here. If you intend to get clcultor pproimtion, your life will e esier if you use common log or nturl log. 3. Use the 3rd Lw of Logrithms to ring the eponent down in front. This is the whole point of using logrithms it gets the eponent on ground level where you cn do ordinry lger to it. 4. Use ordinry lger to solve for the eponent. Emple: A cteri colony doules every 0 minutes. It strts with 3 million cteri t noon. When will there e 8 million cteri in the colony? Solution: If t is in hours nd A(t) is in millions of cteri, the function tht tells how mny cteri in the 3 colony is At 3 t. (Review on your own if you don t rememer how to find this function.) So the eqution we wnt to solve is 8 3 3t First, we do s much ordinry lger s possile to isolte the eponent. For this emple, tht mens dividing oth sides of the eqution y 3: 8 3 3t Tht s s much s we cn do without logrithms. Now it s time to tke the log of oth sides. I wnt clcultor pproimtion when I m done here (so I cn write down time), so I ll use nturl log. You cn use ny log you like, s long s you do the sme thing to oth sides. 8 3t ln ln 3t ln 3 Tking the log rings the eponent down in front (third Lw of Logrithms), which is just wht we wnt. Now we hve n eqution of the form numer = numer times t; it s time to do ordinry lger gin to solve for t. Divide oth sides of the eqution y 3 ln to get ln 8/ 3 t 3ln This is the ect nswer. I cn t just look t this nswer nd see how ig it is, though, so I wnt clcultor nswer. ln 8/ 3 t 3ln This tells me tht the colony will hve 8 million cteri out 0.47 hours, or it more thn 8 minutes, pst noon. Does this mke sense? By counting up we cn see tht the colony would hve 6 million cteri t :0 nd million t :40, so this is resonle. There will e 8 million cteri t out :8.

8 Chpter Review Business Clculus 8 Section : Wht is Function? Functions The notion of function is one of the most powerful in mthemtics. It s surprisingly simple ide, though. The reson students re so often confused when they encounter functions for the first time in n lger clss is the nottion. Before we get to the nottion, we ll concentrte on the core ide. Our lives re full of reltionships nd correspondences etween sets, lthough we don t lwys think of them in these terms. For emple, we know tht the numer of pltes we tke out of the cupord corresponds to the numer of people we re epecting t the tle. We know tht ech telephone numer we know corresponds to one person tht we wnt to rech. We know tht the size of our electric ill corresponds to the mount of electricity we use. A function is just specil type of correspondence. Definition: A function is correspondence etween two sets tht ssigns to ech element of the first set ectly one element of the second set. The first set, the set of inputs, is clled the domin. The second set, the set of outputs, is clled the rnge. Functions do not hve to hve nything to do with numers. The key point is those words ectly one. Tht mkes them predictle, nd tht s the reson they re so importnt. Emple: Every person hs irthdy. This is n emple of function. Notice tht ech person gets ectly one irthdy. Notice lso tht lots of people cn hve the sme irthdy tht doesn t ffect whether this reltionship is function or not. The ectly one only needs to work the one direction. In this emple, the domin is the set of ll people, nd the rnge is the set of ll possile irthdys (the dys of the yer). Emple: Every numer hs squre. This is lso n emple of function. Agin, notice tht every numer hs ectly one squre if you give me numer, I cn give you its squre ( function is predictle). In this cse, the domin is the set of ll numers, nd the rnge is the set of ll possile squres. The point of function is to e predictle, so it s nicest if we cn write down rule. There re severl different wys to write function: A function could come s tle. The income t tles in the ck of the t ooklet re emples of this kind of function. There s one such function every yer for ech type of tpyer: single, mrried filing jointly, etc. Within ech of these tles, the ssignment of t mount to tle income mount is the function, nd the informtion comes from tle. In this emple, the domin is the set of possile tle income mounts nd the rnge is the set of possile t mounts.

9 Chpter Review Business Clculus 9 To tell if tle represents function, you need to check whether ny input hs two outputs. Rememer, function ssocites ectly one output to ech input. In our income t emple, you cn tell it s function ecuse no mtter how mny times you look it up, the mount you owe the government doesn t chnge. Notice tht it doesn t mtter tht severl tle income mounts yield the sme t mount it s OK for mny different inputs to give the sme output. A function could come s grph. For emple, the grph tht shows the Dow Jones verge in the newspper represents function. The domin is long the horizontl is (in my newspper, tht represents the set of the lst five usiness dys), nd the rnge is represented verticlly (the Dow Jones verge for tht dy). The informtion out this function comes from the grph. In order to find the Dow Jones verge for lst Fridy, sy, you red the grph. Every time you red this week s grph for lst Fridy, you ll see the sme Dow Jones verge the grph is predictle. To tell if grph represents function, you need to check whether ny input (long the horizontl is) hs two outputs (vlues ove or elow it on the grph). An esy wy to tell is to use the verticl line test. If ny verticl line hits the grph more thn once, then the grph does not represent function. Emple: The grph of circle is not function, ecuse there re lots of verticl lines tht cross the circle more thn once. This grph fils the verticl line test. The grph of the top hlf of circle is function. A function could come s n lgeric rule. This is the wy most students think out functions (which my e why so mny people ecome confused out functions). This is gret shorthnd wy to write function tht hs to do with numers. For emple, our squre numer emple from ove could e written this wy: f This is red f of equls squred. The f here is the nme of the function. You ll often see f used for function, ecuse f is the first letter in the word function. But ny letter or comintion of letters would e fine. In fct, it s good ide to pick letter tht will remind you of wht you re doing. The prentheses here do not denote multipliction. They re red loud s of. The fct tht they re right net to the nme of the function tells you tht this is function, nd you should look inside them to see wht the vrile will e. The here is the vrile nme. Agin, is very commonly used, ut there s nothing mgic out it. You could use ny letter or symol tht you like. The point is to look within the prentheses to see wht letter is there, ecuse tht s wht will stnd for the input in the rule.

10 Chpter Review Business Clculus 0 The lgeric stuff on the right hnd side of the equls sign is the rule. This is the prt tht tells you wht to do with your input. Your input goes ectly in plce of the vrile (which you identified right ove). This rule sys tke the input nd squre it. Emple: In the function C F 3 5 9, the function nme is C, the vrile nme is F, nd the rule sys first sutrct 3 from your input, then multiply the result y 5/9. This is the lgeric representtion of the function tht ssocites degrees Celsius to degrees Fhrenheit. The domin here is the set of ll possile tempertures, mesured in degrees Fhrenheit, nd the rnge is the set of ll possile tempertures, mesured in degrees Celsius. This is function, ecuse there is ectly one Celsius mesurement corresponding to ech Fhrenheit mesurement. One convenient thing out hving n lgeric representtion for function is tht you don t hve to check whether the ectly one condition is stisfied. Alger hs tht property uilt in you lwys get the sme nswer when you plug in the sme input. The Rule of Four There re four wys tht mthemticl informtion cn e communicted to you. Numericlly- s list of numers in tle, for emple. Algericlly or nlyticlly - s formul. Grphiclly or geometriclly, s grph or picture. In English - the story or word prolem. Ech of these wys hs distinct dvntges nd disdvntges. Depending on wht kind of mthemticl informtion you need to communicte, you might choose just one of these wys, or some comintion of these wys. Mny students re most fmilir with lger nd formuls. And mny mth tetooks seem to focus on formuls. But ll of these wys of looking t mthemticl informtion re importnt. We'll e communicting mthemtics in ll four wys during this course. Numericlly Advntges You get precise informtion - ctul numers. This is often how rel-world informtion comes to you, s numericl dt tht s een collected.

11 Chpter Review Business Clculus Disdvntges There's no informtion out nything tht isn't lredy on your list. Ptterns nd trends re difficult or impossile to find. Algericlly or Anlyticlly (with formuls) Advntges You get precise informtion - you cn solve for n ctul numer. You cn use formul to predict informtion out ny numer you're interested in. Ptterns in the sitution my e reveled y wht we know out the formul. Disdvntges Trends my e difficult or impossile to find. Formuls re mthemticl models only -- the rel world is usully not s net nd tidy s the formul suggests. Communicting Grphiclly or Geometriclly Advntges You get ig picture informtion - you cn esily see trends, chnge, nd growth. You cn esily pproimte the interesting points on the curve. It s quick wy to see wht s relly going on. Disdvntges You cn only pproimte numers, ecept for certin known nd leled points. Communicting in English Advntges This is how rel-world prolems come. Noody outside of mth clss will ever sk you to solve qudrtic eqution. Insted, they'll sk you how mny pounds of slmon you'll need to feed dinner prty of eight, or how much is their shre of the phone ill, or wht's the most efficient speed for running the mchinery on the fctory floor. Disdvntges You usully need to use one of the other wys to solve such prolem. Trnsltion cn e difficult. English is fluid lnguge with mny menings. Sometimes there re legitimte ut contrdictory interprettions of the sme English sttements.

12 Chpter Review Business Clculus Section 3: Lirry of Functions There re few functions tht you should e completely fmilir with. By this time, you should hve seen liner, qudrtic, nd eponentil functions mny times. Liner Functions Liner functions re the simplest kind of functions to work with. Mny reltionships re truly liner, nd mny more cn e pproimted well enough with liner function. Liner functions hve mny helpful fetures their grphs re stright lines, which we know lot out. Their rtes of chnge re simply slopes, which we know how to find. Recognize liner growth, no mtter how the informtion is given to you. Rememer tht there re four wys quntittive informtion cn e presented. Numericlly: Liner functions hve constnt chnge in y for every constnt chnge in. This reflects the grphicl ide of liner function the chnge in y over the chnge in, or Δy/Δ, is the constnt slope of the line. One wy to recognize line is if you see constnt slope in tle of numers. Algericlly: The formul for liner function cn lwys e lgericlly mneuvered into one of the common forms given ove. You cn recognize tht function is liner if it hs only one independent vrile, which is rised to the first power only (no squres, no one-overs, no roots), nd some constnts. Grphiclly: Liner functions re the ones whose grphs re stright lines. In English: Liner functions hve constnt rte of chnge. You cn often recognize the slope y its units; look for frctionl units, rise/run units, like miles per hour, or dollrs per pound, or people per yer. The y- intercept is like the fied cost or the overhed how much y you hve when is zero. Qudrtic Functions Qudrtic functions hve lots of pplictions (for emple, the height of sell cn e modeled with qudrtic function). We lredy discussed how to solve qudrtic equtions. Numericlly: The est wy to tell if tle displys qudrtic function is to grph it. Algericlly: Qudrtic functions cn lwys e lgericlly mneuvered to look like f c. You cn recognize tht function is qudrtic if there is only one independent vrile, nd the only powers you see re nd.

13 Chpter Review Business Clculus 3 Grphiclly: The grph of qudrtic function is prol, sort of curvy V-shpe. The formul cn tell you lot of informtion out the grph: The sign of tells you if the grph opens up ( > 0) or down ( < 0) The y-intercept (where = 0) is t y = c. The solutions of the eqution f c 0 re the zeros, the roots of the function these re the -intercepts of the prol. The verte formul tells you where the high or low point of the prol is: ; y = plug it in. You cn use the sme kind of informtion to go from the grph to formul use the zeros of the function to find the fctors, djust the leding coefficient using the y-intercept. In English: If the function is qudrtic, they ll need to sy so specificlly. One of the most common pplictions is the height of flling ody. Eponentil Functions Eponentil functions re very common. For emple, the compound interest formul is n eponentil function. And mny nturl things grow (or decy) eponentilly. Numericlly: Eponentil functions show constnt rtio in y for constnt chnge in. Tht is, if you increse y, you multiply y y constnt multiplier. The multiplier is the esiest se to use for the eponentil function. f 0 A Algericlly: An eponentil function is of the form, where > 0 nd. The domin of the eponentil function is the set of ll rel numers we cn use ny rel numer s the eponent. The rnge of the eponentil function is the set of ll positive rel numers. (If we rise positive numer to ny power, we get positive numer ck.) Note some ooks re totlly e-hppy. Tht is, they wnt every eponentil function to hve se e. Now, e is lovely numer, nd it s the perfect se for some pplictions for emple, continuously compounded interest. But it s etter ide to let e the multiplier. Tht is, if you hve quntity tht doules every hour, you ll e much hppier if you use =. Emple: Suppose cteri colony is growing in such wy tht it doules in size every 0 minutes. There re 3 million cteri t noon.. How mny will there e t :00 pm?. How mny will there e t :30 pm?

14 Chpter Review Business Clculus 4 Solution: Becuse the douling time is constnt, we know the cteri re growing eponentilly.. This prt is esy to figure out without writing formul, y just counting up. If they doule every 0 minutes, then there re 6 million t :0, there re million t :40, nd there re 4 million t :00.. This prt is not so esy :30 isn t whole numer of 0-minute chunks fter noon. So we will uild the formul. Our units will e millions of cteri nd hours. The initil mount, the principl, A 0 is the 3 million cteri we strted with t noon. Our popultion is douling every twenty minutes, so it s eing multiplied y every /3 hour. Over one hour, then, it will e multiplied y 3. The formul tht tells how mny million cteri there re in this colony t hours fter noon is 3t t 3. A 3.5 :30 is t =.5 hours pst noon, so there should e A million cteri. Does this mke sense? Yes, y counting up we find tht there should e 48 million t :0 nd 96 million t :40, so this seems right. Note: You cn pick whtever units re convenient for you. Your formul my end up looking different, ut your nswers will e correct. In the cteri emple, you could hve used the units of (single) cterium A t 3,000,000 nd twenty-minute-intervls. Then the formul would look different: t, ut you d use t = 4.5 (ecuse :30 is 4.5 twenty-minute-intervls pst noon) nd you d get the sme nswer out million cteri. Grphiclly: If >, f represents eponentil growth, nd the grph of the function will e incredily flt on the left, incredily steep on the right. If 0 < <, f represents eponentil decy, nd the grph of the function will e the mirror imge, left to right, of n eponentil growth grph. It will e incredily steep on the left nd incredily flt on the right. Figure We lwys get two points for free on ny simple eponentil grph: (0, ) nd (, ).

15 Chpter Review Business Clculus 5 In English: Eponentil functions show up when the increse depends on how much is lredy there. For emple, compound interest (the dditionl interest depends on how much is in the ccount), or simple popultion growth (the numer of dditionl ies depends on how mny people re in the ccount). Other functions There re severl other functions tht you should know something out you should recognize their formuls nd their grphs. You should know the solute vlue functions polynomil grphs in generl, cuics (3 rd degree) in prticulr rtionl functions (rememer verticl symptotes?) power functions ( n f for some n), including the squre root function logrithmic functions with se 0 nd the nturl log, with se e Section 4: New Functions from Old Trnsformtions Chnging the constnts tht pper in n lgeric formul chnges the grph in some predictle wys. Here re the principles: Chnging the (the input) chnges the horizontl. Chnging the y (the output) chnges the verticl. Multiplying y constnt stretches (or squshes) the grph. Multiplying y reflects the grph. Adding constnt shifts the grph. Here re the detils: Strt with the grph of y = f(). The grph of ech of the following will hve the sme sic shpe s y = f(), ltered s noted. For ll of these, is constnt y = f() is times s tll. y = f() is reflected verticlly cross the -is (upside down). y = f() + is shifted up units. y = f() is times s wide. y = f( ) is reflected horizontlly cross the y-is. y = f( + ) is shifted to the left units. Notice tht for horizontl stretches or shifts, the effect is sort of ckwrds from wht you might epect t first. These cn e confusing check your nswers y plotting couple of points.

16 Chpter Review Business Clculus 6 You cn hndle these ll t once. You cn do horizontl nd verticl chnges independently. For ech of these, follow the order of opertions stretch nd reflect first, then shift. Use the origin s your nchor point, even if it s not on your grph. Emple: The grph of y = hs the sme sic shpe s y =. It s een shifted to the right units. It s upside down, 4 times s tll, nd hs een shifted up 5 units. So the new grph hs its verte t (, 5), opens down, nd looks stretched verticlly compred to the originl. You cn plug in point or two to confirm your nswer. The new grph goes through the point (0, 3), which mkes sense. If you move units to the left of the verte, the unchnged grph goes up units. Here, it goes down (ecuse the grph is upside down) 8 units (ecuse the grph is stretched y fctor of 4.) Figure Composition One of the most importnt wys to comine functions is to chin them together, using the output from one s the input into nother. A simple emple of this is unit conversion we hve one function tht tells us how mny meters high the ll is fter t seconds, nd nother tht tells how mny feet re in certin numer of meters. We cn use the output of the first function (meters) s the input to the second function to find how mny feet high the ll is fter t seconds. The chining together of functions in this wy is clled composition: The composition of f with g, written output. Tht is, f g f g. f g, is the function tht tkes, first does g to it, nd then does f to the

17 Chpter Review Business Clculus 7 Emple: Let f 4 3nd g f g f g f Then their composition Decomposition Sometimes you will e given the composition nd e sked to identify the component pieces. This is clled decomposition. It turns out to e very useful skill in clculus. There re often severl correct decompositions for function, ut usully only one of them is useful. It my tke some prctice efore you cn see which composition is the useful one. Emple: The function G is composition f g. Identify the component functions f nd g. 3 The most useful solution: In mny cses, there is n ovious choice, which you cn find y thinking out the inside nd the outside. In G, the inside function is the denomintor nd the 3 outside function is the reciprocl function (tht is, one over ). In the composition f g, g is the inside function nd f is the outside function. So this decomposition would e f nd g 3. Then F f g f 3. This is the most useful solution. This is the solution you would 3 see in the nswer pges of the tetook. This is the type of decomposition you should look for. There re usully lots of correct solutions, some of which involve some cretivity to find. In this clss, you don t hve to ever find ny of these clever decompositions. If you do find one, it will e correct. But your techer my suggest tht you stick to the more useful decomposition. Inverse Functions The word inverse mens ckwrds, nd tht s wht inverse functions re out going ckwrds. There re few different nd useful wys to think out inverse functions. Swpping the roles of input nd output One importnt reson we cre out inverse functions is tht, in mny cses, the sme reltionship cn give two different functions, depending on wht questions you re interested in nswering. Which function you use depends on which quntity you wnt to use s your input.

18 Chpter Review Business Clculus 8 Emple: A privte investigtor chrges $500 fee per cse, plus $80 per hour tht she works on the cse. There is functionl reltionship etween the hours she works nd the mount she ills. But which is the input nd which is the output? If the numer of hours she works is the input, then the numer of dollrs she ills is the output. And it s function, ecuse ech possile numer of hours is ssocited with ectly one illing mount. This might e the function you d think of first. If we let h e the numer of hours the detective works nd e the numer of dollrs she ills, then this function might e written s f h h. You d use this function if you knew how mny hours she worked on your cse nd you wnted to know how much she would chrge you. But the very sme reltionship yields different function, whose input is the illing mount nd whose output is the numer of hours she works. This is lso function, ecuse ech possile ill is ssocited with ectly one mount of time. Agin, letting h e the numer of hours nd e the mount she ills in dollrs, we cn write this function: h f This would e helpful function if you hd certin mount of money to spend nd you wnted to know how mny hours she would work on your cse. The two functions here re inverse functions. They model the sme reltionship, ut the roles of input nd output hve een echnged. Tht little tht looks like n eponent for the f in the second formul indictes it is the inverse function for f. (It is not n eponent.) Undoing The most importnt reson we wnt to study inverse functions is tht they undo ech other. Rememer the lgeric definition of inverse functions: f nd f oth re inverse functions mens tht their composition in either order is the identity function. Tht is, f f nd f f The rrow digrm my e clerer:

19 Chpter Review Business Clculus 9 f f f f f nd f Grphiclly If you grph function nd its inverse on the sme es, the inverse will e reflection of the originl cross the line y =. Tht s ecuse the inverse function swps the roles of input nd output. On grph, tht mens interchnging the order of the coordintes for every point. Tht is, if (, y) is on the grph of y f, then (y, ) will e on the grph of y f. Chpter Eercises. Use the rules of eponents to simplify the following. Write your nswer using only positive eponents. Assume ll vriles represent non-zero numers.. 4y y y z 3 c. 8 m 3 5 yz e. 5 7 d. 0 f Write the eqution of the line tht is prllel to y.5 nd hs y-intercept t y = Write the eqution of the line tht psses through the points (, 5) nd (3, ). 4. A coffee supplier finds tht it costs $850 to rost nd pckge 00 pounds of coffee in dy nd $850 to produce 500 pounds of coffee in dy. Assume tht the cost function is liner. Epress the cost s function of the numer of pounds of coffee they produce ech dy Fctor y 6y 9y. 6. Fctor The two points, 8 3 nd,0 oth lie on the grph of function g. Find the formul for y g if it is liner function.. Find the formul for y g if it is of the form C g. y. 8. An oject is thrown into the ir. Its height in feet ove the ground t seconds lter is given y h t 6t 30t 5.. Find the time when the oject reches its mimum height.. How high does it get? c. When does it lnd?

20 Chpter Review Business Clculus 0 d. The grph of y = h(t) is prol. Wht is the physicl mening of the y-intercept of this prol? (Tht is, wht does the y-intercept of the prol tell us out the oject?) 9. Suppose u v nd v u. Find possile formuls for u nd 0. Suppose f. f f 3. Compute. Simplify your nswer. 3. In English, wht is the mening of your nswer to prt? c. Grphiclly, wht is the mening of your nswer to prt? v.. Here is the grph of y f. Use this grph to sketch the grph on grph pper of y 3 f creful nd net, nd rememer to lel your grph. Briefly eplin wht you did to find the grph.. Be Figure 3. In 000, the numer of people infected y virus ws P 0. Due to new vccine, the numer of infected people hs decresed y 4% ech yer since Find formul for P f n, the numer of infected people n yers fter When will there e (or when were there) just hlf s mny people infected s there were in 000? 3. Here is the grph of n eponentil or logrithmic function.

21 Chpter Review Business Clculus Figure 4. Is this n eponentil function with se >, n eponentil function with 0 < <, or logrithmic function with se >?. Wht is the vlue of (the se) for this grph? How do you know?