Algebra I – Functions and Variations - Quick Guides | Mathematics

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Study GuideAlgebra I–FuncƟons and VariaƟons1. FuncƟons1. Understanding Relations FirstBefore we learn aboutfunctions, we need to understand something called arelation.Arelationis simply a set of ordered pairs.Anordered pairlooks like this:[(x, y)]The first number is thex-value, and the second number is they-value.So, if we have:[A = {(-1,1), (1,3), (2,2), (3,4)}]This set of ordered pairs is called arelation.“Figure 1shows a graph of the ordered pairs (-1,1), (1,3), (2,2), (3,4).”

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Study Guide2. Domain and RangeNow let’s talk about two very important ideas:DomainThedomainis the set of all x-values.RangeTherangeis the set of all y-values.From the example:[A = {(-1,1), (1,3), (2,2), (3,4)}]•Domain = {−1, 1, 2, 3}•Range = {1, 2, 3, 4}Notice:•We list each number only once.•Order does not matter in sets.ExampleFind the domain and range of the plotted points.Domain = {−2,−1, 1, 3}Range = {−1, 2, 3}

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Study Guide“Figure 2Plotted points.”3. What Makes a Relation a Function?Now we move to something more specific:functions.Afunctionis a special type of relation.Definition of a FunctionA function is a relation where:Each x-value is paired with exactly one y-value.That means:•No x-value can have two different y-values.•Each input gives only one output.4. Example of a FunctionConsider the equation:[y = x + 1]

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Study GuideThis is a function because:•For every value of x,•There is exactly one value of y.For example:•If x = 2, then y = 3•If x =−1, then y = 0There is never more than one answer for y.The domain and range of this function are:•All real numbers“Figure 3.A graph of the linear equation y = x + 1.”5. Graphs of FunctionsLet’s look at some more examples of functions.(a)(y = |x|)

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Study GuideThis graph forms a V-shape.Each x-value gives exactly one y-value.(b)(y = x2)This graph is a parabola.Again, every x has only one y.(c)(y = sin x)This graph is a wave pattern.Still, each x corresponds to one y-value.All of these are functions.“Figure 4Graphs of functions.”6. Graphs That Are NOT FunctionsNow let’s see what isnota function.A relation isnot a functionif:A single x-value is paired with two or more y-values.

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Study GuideLet’s examine these cases:(a) x = 2This is a vertical line.The same x-value (2) has many y-values.Not a function.(b) (x2+ y2= 4)This is a circle.Some x-values match with two y-values.Not a function.(c) (x = y2)This sideways parabola gives two y-values for some x-values.Not a function.•x = 2•circle (x2+ y2= 4)•(x = y2)

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Study Guide“Figure 5Graphs of relations that are not functions.”Example 2:Understanding Domain, Range, and FunctionsWhat Is a Relation?Arelationis simply a set of ordered pairs.For example:[B = {(-2,3), (-1,4), (0,5), (1,-3)}]Each ordered pair has:•Afirst number→this belongs to thedomain•Asecond number→this belongs to therangeWhat Is the Domain?Thedomainis the set of all x-values (first numbers).From set B:Domain = {−2,−1, 0, 1}Think of the domain as:

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Study Guide“What values go INTO the function?”What Is the Range?Therangeis the set of all y-values (second numbers).From set B:Range = {−3, 3, 4, 5}Think of the range as:“What values come OUT of the function?”When Is a Relation a Function?A relation is afunctionif:Each x-value is paired withonly oney-value.If one x-value has two different y-values, it isnota function.Looking at GraphsNow let’s understand what the graphs are showing.1 & 2—Checking Domain and RangeYou are given sets and asked to:•Identify thedomain•Identify therange•Decide if it is afunction

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Study GuideIf each x appears only once→FunctionIf an x repeats with different y-values→Not a function3—Graph with Several PointsLook at the plotted points carefully.To find the:•Domain→look at all x-coordinates•Range→look at all y-coordinatesTo check if it’s a function:Use theVertical Line Test.If a vertical line touches the graph at more than one point→Not a function.

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Study Guide4—Range Given as an InequalityRange: { y : y≥−3 }This means:All y-values are greater than or equal to−3.If every x gives exactly one y→it is a function.5—A Line with an ArrowThe graph shows:•A solid point at (0,0)•A line extending upward•A point at (1,−3)Domain is written as:

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