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Algebra II - Relations and Functions

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Study GuideAlgebra II–RelaƟons and FuncƟons1. Basic DefiniƟonsIn algebra, a concept can sometimes have more than one way to describe it. This is true fororderedpairsandrelations.Anordered pairis written as ((x, y)).•Thefirst number (x)represents thehorizontal coordinate.•Thesecond number (y)represents thevertical coordinate.When several ordered pairs are grouped together, they form arelation.•The set of allx-valuesin a relation is called thedomain.•The set of ally-valuesin a relation is called therange.A special type of relation is called afunction. In a function,each x-value is paired with only one y-value.Relation

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Study GuideFigure 1 Examples of relations.Arelationis simply aset of ordered pairs.Relations can be shown in different ways, such as:•Alist of ordered pairs•Amapping diagram•Agraph•AnequationEach of these representations shows how values in the domain relate to values in the range.Domain and RangeThedomainof a relation is the set of allfirst numbers (x-values)in the ordered pairs.Therangeis the set of allsecond numbers (y-values).When listing domain and range, the numbers are usually writtenfrom smallest to largest.

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Study GuideExample 1:FindFind thedomain and rangeof item (a) in Figure 1.Ordered pairs:Domain:The x-values are (4, 3,) and (-1).Range:The y-values are (1, 2,) and (6).Example 2:FindFind thedomain and rangeof item (b) in Figure 1.From the mapping diagram:•Domain values: (1, 2, 3)•Range values: (4, 5)Example 3:FindFind thedomain and rangeof item (c) in Figure 1.Domain (x-values):

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Study GuideRange (y-values):Example 4:FindFigure 2The domain is horizontal.Figure3 The range is vertical.

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Study GuideFigure 4 This is not a function.Figure 5. Like Figure 4, this is also not a function.

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Study GuideFigure 6. Functions have no repeatedxvalues.Figure 7. This fails the vertical line test and is, therefore, not a function.Find thedomain and rangeof item (d) in Figure 1.In this example, the values cannot simply be listed. Instead, we describe them usinginequalities.Finding the Domain

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Study GuideImagine moving each point on the graphstraight down to the x-axis.The points on the x-axis represent the domain.The domain includes all (x) values from-3 to 4, but-3 is not included.This is read as:“the set of (x) values such that (x) is greater than-3 and less than or equal to 4.”Finding the RangeNow imagine sliding each pointhorizontally to the y-axis.These points show therange.The y-values go from-1 to 2.This means:“the set of (y) values such that (y) is greater than or equal to-1 and less than or equal to 2.”Example 5:FindFind thedomain and rangeof item (e) in Figure 1.Equation:For this linear equation:•Any value of (x) can be used.•Each value of (x) produces a value of (y).So,

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Study GuideFunctionAfunctionis a special type of relation.In a function,each value in the domain is paired with exactly one value in the range.This means:•A singlex-value cannot match with two different y-values.Example 6:Identifying FunctionsLook at the examples from Figure 1.Example (a)Thisis a functionbecauseno x-value is repeated.Example (b)The mapping diagram shows an input connected tomore than one output.Therefore,it is not a function.Example (c)In this graph, the vertical line (x=-1) would pass throughtwo points.This means the graph fails thevertical line test, so it isnot a function.The Vertical Line TestThevertical line testhelps determine whether a graph represents a function.Rule:If a vertical line crosses the graphmore than once, the relationis not a function.If every vertical line touches the graphat most once, the relationis a function.

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Study GuideExample: (y = 2x + 3)This equationis a function.When the graph of (y = 2x + 3) is drawn:•Every vertical line intersects the graphonly once.So, itpasses the vertical line test.Key Ideas to Remember•Arelationis a set of ordered pairs.•Thedomainis the set of allx-values.•Therangeis the set of ally-values.•Afunctionis a relation where each input hasonly one output.•Thevertical line testhelps determine whether a graph represents a function.SummaryArelationconnects values using ordered pairs ((x,y)). Thedomainis the set of all x-values, and therangeis the set of all y-values. Relations can be shown using ordered pairs, mapping diagrams,graphs, or equations.Afunctionis a special relation in which each x-value corresponds to exactly one y-value. One way tocheck if a graph represents a function is by using thevertical line test. If any vertical line intersectsthe graph more than once, the graph does not represent a function.2. Quiz: Basic DefiniƟons1. QuestionA relation contains the points (-10, 5), (-5, 1), (0,−3). What is therangeof that relation?

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Study GuideAnswer Choices•(−3, 0, 5)•(−10,−5, 0)•(−3, 1, 5)Correct Answer(−3, 1, 5)Why This Is CorrectTherangeof a relation is the set of ally-values.From the ordered pairs:(-10,5)(-5,1)(0,−3)The y-values are:5, 1,−3So the range is:{−3, 1, 5}2. QuestionA relation contains the points {(3, 2), (−2, 4), (5,−3)}. What is thedomainof that relation?Answer Choices•(−2, 3, 5)•(−3, 2, 4)•(−3,−2, 2, 3, 4, 5)
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Document Details

University
Texas A&M University
Course
Algebra II
Subject
Mathematics

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