1.1 Functions, Domain, & Range - Notes [Teacher]

[1.1] FUNCTIONS, DOMAIN, & RANGEWHAT IS A RELATION?A "relation" is just a relationship between sets of information.Think of all the people in one of your classes and think of their heights. The pairing ofnames and heights is a relation.In relations and functions, the pairs of names and heights are "ordered", which meansone comes first and the other comes second.To put it another way, we could set up this pairing so that either you give me a name,and then I give you that person's height, or else you give me a height, and I give you thenames of all the people who are that tall.The set of all the starting points is called "the domain" and the set of all the endingpoints is called "the range." The domain is what you start with; the range is what youend up with. The domain is the x's; the range is the y's.WHAT IS A FUNCTION?A function is a "well-behaved" relation.Just as with members of your own family, some members of the family of pairingrelationships are better behaved than other. Warning!! This means that, while allfunctions are relations, since they pair information, not all relations are functions.Functions are a sub-classification of relations.When we say that a function is "a well-behaved relation", we mean that, given a startingpoint, we know exactly where to go; given an X, we get only and exactly one y.WHAT IS THE DIFFERENCE BETWEEN A RELATION AND A FUNCTION?RELATIONFUNCTION1. A relation is defined as a relation1. A function is defined as a relation inbetween sets of values. Or, it is a subset ofwhich there is only 1 output for eachthe Cartesian product.input.2. A relation is denoted by "y="2. A function is denoted by "f(x)"3. Every relation is NOT a function3. Every function is a relation.1 [1.1] FUNCTIONS, DOMAIN, & RANGEWHAT IS A VERTICAL LINE TEST (V.L.T.)?The vertical line test is a method that is used to determine whether a given relation isa function or not. The approach is rather simple.Draw a vertical line cutting through the graph of the relation, and then observe thepoints of intersection.The vertical line test supports the definition of a function. That is, every x-value ofafunction must be paired to a single y-value.If we think of a vertical line as an infinite set of x-values, then intersecting the graph ofa relation at exactly one point by a vertical line implies that a single x-value is onlypaired to a unique value of y.PASSES V.L.T. IS A FUNCTIONLinear: f(x)=x+1Quadratic: f(x) = x2 - 2yxone point ofintersection\one point ofintersectionCubic: f(x)=one pointintersectionFAILS THE V.L.T. IS NOT A FUNCTIONSideways Parabola: x = y2Circle: x2 +y2=9yintersecting attwo pointsxXintersecting attwo points2 [1.1] FUNCTIONS, DOMAIN, & RANGEReverse Cubic: x = y3 -y+2 =yintersecting atthree pointsXWHAT IS A DOMAIN?The domain is the set of all X or input values. We may describe it as the collectionofthe first values in the ordered pairs.WHAT IS A RANGE?The range is the set of all y or output values. We may describe it as the collection ofthe second values in the ordered pairs.ORDERED PAIRSTABLE OF VALUESGRAPHMAPPING DIAGRAMy4xy3-2-3-212;01-23113o-3-1-1-23414listof-331+XY allcoordinuteSET NOTATION = E(-2,1), (-2,3), (0,-3), (1,4), (3,1)34 pointsDOMAIN = E-2,0,1,3}/Get rid ofduplicatesRANGE = ¿-3,1,3,4}2write them inincreasing order+-3 [1.1] FUNCTIONS, DOMAIN, & RANGEDETERMINE IF THE FOLLOWING IS A FUNCTION OR NOTEXAMPLESFUNCTION? EXPLAIN WHY OR WHY NOTIDuplicates of x-valves - with different y-valuesa. (-3,7)(-1,5) (0,2),59)53)2Not a Functionb./xyEvery x value is unique and-2oassociated with any 1 y-value-1-2O32IS a Function4-1=5-3C.X4/There is repeating values of 11/1"xy15x5but they are associated with15same "y" value.383-82Everyx valve is unique and3-83-83 IS a function associated withing 1 y-valued.domainrangeIConvert to Coordinates:5o( (5,0) (7,1) (8,1) (10,2)712Every x value is associated with 1 y-valee82103IS a Function=e.IConvert to Coordinates:domainrangeCl, -2) (2,-2) (3,-2) (4,-2)122Domain values am share a common value-2in rage4IS a Funch's343Every x value is associated with 1 yvaluef.ICenvert to Coordinatesdomainrange-65 -6.) (5)-2) 5 4) (5,11)5-22Duplicates of x-values with different yevalus4113NOT a Function4 [1.1] FUNCTIONS, DOMAIN, & RANGEg./Convert to Coordinatesdomainrange(1 1,5) 2? (3,-3) (4,14)1-3252 have an output in the rangeno pair! Every ver (ve in Domain most231443This is NOT a Function or a RelationPROVIDE THE DOMAIN & RANGE FOR THE FOLLOWING FUNCTIONSALWAYSDomain and RangeAskYOURSELFRANGE"ARE THEREDOMAINANY RESTRICTIONDomain is all the possibleRange is all the possible yx values of a function.values of a function.FUNCTIONDOMAIN AND RANGE1. LinearSolutions.yTIm"x" can be anyD =real number2,range: all y-valuesR = { YER"y" can be anyreal numberLD Both mean infinite solutionsdomain: all x-valueswith no restrictions2. QuadraticD =EXER}No restrictionsy=2x2Range: ys 2R =Vertex = (o 2y=2aRestrictionDomain: all x-valuesEYER / yszz11g 11 can be any real number when5it is less than or equal to 42" [1.1] FUNCTIONS, DOMAIN, & RANGE3. Sineyy=1 1D= EXER} No restrictionsi234567-1y=-1R = restriction4. Exponentialg(x)D = EXER}no restrictionsy=21R = { YERI y > }x-3-2123y greater than 2 (never touches-1asymptote, justruns along it!5. RationalX=SD = EXER / x #2}all X- valves, but does not2y=0Rew1include x=2 (asynptote)R.O-Range: y 0vertical2HorizontalR= = w YER/ y=o}AsymptoleAsymptoteall y-values, but does notODomain: Y # 2include y=o (asymptote)6. Radicalx=2yD = EXER I yzz3 restriction43RA2Range:0+R= { YER I yzoz-y=orestrictionDomain:1226