MTH 111 Cllege Algebra Lecture Ntes July 2, 2014 Functin Arithmetic: With nt t much difficulty, we ntice that inputs f functins are numbers, and utputs f functins are numbers. S whatever we can d with numbers, we can d with functins! Fr starters, we can add, subtract, multiply, and divide functins. Fr the dmains f each f the first three, we keep what s cmmn between the dmains f and respectively (frmally, this is the intersectin f the dmain f with the dmain f ). Fr the dmain f the qutient, we start with the intersectin f the tw dmains f and, and then we need t exclude any values which wuld cause the denminatr t be zer. Ntatin:, plus f is the same as, f plus f, is the same as, is the same as, is the same as, is the same as [ ] [ ], and is the same as, is the same as. Cnsider the example: The dmain f and wuld each be the intersectin f the dmains f and. Well, the dmain f is all real numbers because it desn t break any rules and there is n cntext (stry). And the dmain f is all since we can t take the square rt f a negative. What s cmmn between all real numbers and all numbers greater than r equal t 1? We can see frm the graph abve that the intersectin is, i.e., [ ). 1
MTH 111 Cllege Algebra Lecture Ntes July 2, 2014 The dmain f is similar in that we start with the intersectin (in this case ), but then we need t exclude any values that cause divisin by zer. We can see that, s we have t exclude. That is, the dmain fr is, i.e.,. Furthermre, we culd evaluate things like the sum, difference, prduct, and qutient at specific values f. Sme examples fllw. As a basic example, let s first evaluate. Befre we evaluate it thugh, we ll change the ntatin t smething mre palatable, namely. Using the frmulas, we see that and. S. Similarly,,, and. Hwever, things like and are undefined because 0 is nt in the dmain f and 1 is nt in the dmain f. Ntice in the latter case, that when we try t evaluate the qutient, we get At the same time, we shuld nt be verzealus in ur applicatin f these rules. Fr example, There is a prpensity amng students t get verzealus when they see the zer in the numeratr and cnflate the tw ideas f divide by zer and divide int zer. [An IMPORTANT nte: It is cmmn t view the parentheses in functin ntatin as multiplicatin since that is hw we usually use parentheses. Hwever, it is EXTREMELY imprtant t realize that the parentheses in functin ntatin is ONLY ntatinal and has NO arithmetic significance. Hence the emphasis placed n the f in earlier discussins.] In terms f frmulas, we culd (and ften d) write, fr example,, r, r. Check that yu culd find that the dmain f is [ ), using the same tw functins. 2
MTH 111 Cllege Algebra Lecture Ntes July 2, 2014 Fundamental Principle f Graphing: Befre mentining the Fundamental Principle f Graphing, it will be beneficial t nte that any mentin f the wrd fundamental in a mathematics cntext is a clue as t its imprtance. Juuuust sayin. Yur textbk ffers a frmal explanatin f the fundamental principle f graphing, s I ll ffer a mre infrmal ne. The Fundamental Principle f Graphing says that the -value f the crdinate n the graph f a functin is the functin evaluated at that, namely. That is, ( ). We ve already used this principle when evaluating functins using graphs, and that s fairly straightfrward. Hwever, there are ways this principle shws up mre subtly. the bvius use a less bvius use In the graph at the right, nthing is t surprising, but it is smething that is ften verlked by students. We knw that any pint n the -axis has the -crdinate f zer, s wherever the functin crsses the -axis has the -crdinate equal t zer. The way this presents itself in prblems is that we are asked either explicitly r implicitly t slve fr, r wrk with, the values f which slve. We call these special pints n the graph f a functin each f the fllwing interchangeably: -intercepts, Rts, Zers, r Slutins. Anther, VERY IMPORTANT, use is slving equatins. We will revisit this at a later date. 3
MTH 111 Cllege Algebra Lecture Ntes July 2, 2014 Functin Behavir: There are five types f functin behavir that we fcus n here: Intervals n which the functin is Increasing, Intervals n which the functin is Decreasing, Intervals n which the functin is Cnstant, Maximum values f the functin, and Minimum values f the functin. [Nte: In the big picture, we study the first three t infrm us abut the last tw. Sadly thugh, we dn t get t see this until Calculus.] Fr the first three, we adpt the cnventin f reading the graph frm left t right, and ur default answers are intervals alng the -axis. Thugh there are ther ways f reprting ur answers, which will be discussed at the end f this bullet. Cnsult pages 100 & 101 f yur textbk, and nte that we will treat ur answers slightly differently than the textbk treats theirs (nted belw). Ging frm left t right, we imagine urselves walking n the functin. Here we can see that we are walking uphill frm t, then dwnhill frm t, then uphill again frm t, then we are n level grund frm t, then finally uphill frever after. Mathematically, we say that the functin is increasing anywhere the abve paragraph says uphill, decreasing anywhere it says dwnhill, and cnstant anywhere it says level grund. Furthermre, rather than listing all f the frm-t pints, we just reprt the intervals n the - axis which crrespnd t thse pints. S, fr the graph abve, we wuld reprt that the functin is increasing n the intervals and, and the functin is 4
MTH 111 Cllege Algebra Lecture Ntes July 2, 2014 decreasing n the interval, and the functin is cnstant n the interval. [Nte: The textbk uses brackets fr their intervals, but we will use parentheses in this class.] Yu may, if yu wish, reprt yur answers like, increasing frm t and s n. Hwever, yu will need t be careful with the s, and yu ll likely need t write cmplete English sentences! YIKES! I say g with the intervals. Fr the last tw f the five behavirs mentined abve, namely maximum and minimum, we break these up int tw types: Glbal and Lcal (r equivalently, Abslute and Relative). S that means there are Glbal Maxima, Lcal Maxima, Glbal Minima, and Lcal Minima. Taking maxima and minima tgether, we call them extrema, which means there are glbal extrema and lcal extrema. We prefer the wrd extrema when we dn t knw which kinds there are in advance, r when the teacher is trying t avid giving any hints. Fr a frmal discussin n extrema, cnsult pages 101 & 102 f yur textbk. Here are sme useful ways f thinking abut them: A pint is a Glbal Maximum fr a functin if it is abve every ther value f the functin, and it is a Glbal Minimum if it is belw every ther value f the functin. A pint is a Lcal Maximum if there is an pen interval n which the pint lks like a glbal max, and a pint is a Lcal Minimum if there is an pen interval n which the pint lks like a glbal min. Fr lcal extrema, we require pen intervals because f technical stuffs, which we ll get t in a bit. But what des it mean t lk like a glbal [extremum]? We explain this by way f pictures: 5
MTH 111 Cllege Algebra Lecture Ntes July 2, 2014 Here, we can see that the pint ( ) lks glbal if we zmed in and nly lked in the bx. Similarly with the pints. Hwever, nne f these fur pints are actually mre extreme than all ther value f the functin. S they are all lcal extrema, and nne f them are glbal extrema. In shrt, we say that the functin has a lcal maximum at and at, and has a lcal minimum at and at. Additinally, the functin has a glbal minimum at ( abve that pint. ) since the functin is everywhere Nte that the functin has NO glbal maximum because it ges up frever. Nte further, that it wuld seem that the pint culd als be a lcal minimum. Hwever, that thing abut pen interval in the technical definitin is what keeps us frm being able t d that. In effect, if the functin is nt defined n ALL pssible in the prpsed interval, it cannt be a lcal extreme value fr the functin. That means the functin needs t be n bth sides f the extreme value. *** Sme strange phenmena regarding extrema: We allw fr a pint t be bth glbal and lcal. Cnsider the vertex f a parabla. Using the technical definitin in the textbk, all the pints where the functin is cnstant are BOTH maxima and minima simultaneusly! *** Smetimes a functin will have NO extrema. Cnsider a line. Smetimes it ll seem t us that a pint is an extrema, but if the functin is nt defined there (gaps r hles r pen circles, etc.) it is nt an extreme value. Fr example, if we changed the pint abve frm a clsed circle t an pen circle, it wuld cease t be a glbal minimum, and we wuld say that the functin has n glbal minimum. If in dubt, try picking a pint yu think is the glbal minimum. I ll just pick the pint that is 0.01 r 0.001 r 0.0001, r 0.00..01 belw yurs that is n the functin. Since we can keep getting infinitely clse t the pen circle, but never attain it, it is nt an extreme value. [*** = We will mstly avid the nitty gritty fr these this term. Intuitin will suffice.] Why d all f this? Because it is ne f the mre interesting questins we can tackle at this level f mathematics. Fr example, if we have a functin mdeling prfit, we can nw answer the questin, Hw d we make the mst mney pssible? If we have a functin mdeling travel frm A t B, we can answer the questin, Hw d we get there as quickly as pssible? These are just a few uses. 6
MTH 111 Cllege Algebra Lecture Ntes July 2, 2014 Smething t think abut: Using ur cnsiderable expertise regarding lines and linear equatins, we will apprach the very difficult questin belw: If these are tw business mdels fr taxi cab meter rates, which is better in a city like Astria? Nw this questin is nt pssible t answer withut the mathematics, ur experience in life, and assumptins regarding the city f Astria and its inhabitants. Things t think abut: What is the practical meaning f that pint f intersectin? It is where bth mdels have the same price. In mdel 1, hwever, it is mre expensive fr distances exceeding 1.6 miles, while mdel 2 is mre expensive fr distances less than 1.6 miles. S which is better in Astria? Well, that depends n hw we answer what fllws and ther things like them. Shuld we assume the custmer base fr the taxi cab cmpany is entirely ratinal, entirely infrmed, and interested in being either r bth f thse? What is mre likely frm the custmer base in Astria, lng taxi rides r shrt taxi rides? Gegraphically, it is a rural city, which is lng and narrw, s it s a tugh questin. [When cnfrnted with questins like this in this class, yu wn t be expected t cme t the right cnclusin. Hwever, yur cnclusin MUST be infrmed by the mathematics in additin t the life experiences, assumptins, etc. that yu bring t the table. Withut using the mathematics, yur cnclusin will be cnsidered unjustified.] 7