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Algebra II - Relations and Functions

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    Study Guide Algebra II – Rela Ɵ ons and Func Ɵ ons 1. Basic De fi ni Ɵ ons In algebra, a concept can sometimes have more than one way to describe it. This is true for ordered pairs and relations . An ordered pair is written as ((x, y)). • The first number (x) represents the horizontal coordinate . • The second number (y) represents the vertical coordinate . When several ordered pairs are grouped together, they form a relation . • The set of all x - values in a relation is called the domain . • The set of all y - values in a relation is called the range . A special type of relation is called a function . In a function, each x - value is paired with only one y - value . Relation

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    Study Guide Figure 1 Examples of relations. A relation is simply a set of ordered pairs . Relations can be shown in different ways, such as: • A list of ordered pairs • A mapping diagram • A graph • An equation Each of these representations shows how values in the domain relate to values in the range. Domain and Range The domain of a relation is the set of all first numbers (x - values) in the ordered pairs. The range is the set of all second numbers (y - values) . When listing domain and range, the numbers are usually written from smallest to largest .

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    Study Guide Example 1: Find Find the domain and range of 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: Find Find the domain and range of item (b) in Figure 1. From the mapping diagram: • Domain values: (1, 2, 3) • Range values: (4, 5) Example 3 : Find Find the domain and range of item (c) in Figure 1. Domain (x - values):

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    Study Guide Range (y - values): Example 4 : Find Figure 2 The domain is horizontal. Figure 3 The range is vertical.

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

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    Study Guide Figure 6. Functions have no repeated x values. Figure 7. This fails the vertical line test and is, therefore, not a function. Find the domain and range of item (d) in Figure 1. In this example, the values cannot simply be listed. Instead, we describe them using inequalities . Finding the Domain

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    Study Guide Imagine moving each point on the graph straight 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 Range Now imagine sliding each point horizontally to the y - axis . These points show the range . 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 : Find Find the domain and range of 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 Guide Function A function is 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 single x - value cannot match with two different y - values . Example 6: Identifying Functions Look at the examples from Figure 1. Example (a) This is a function because no x - value is repeated . Example (b) The mapping diagram shows an input connected to more than one output . Therefore, it is not a function . Example (c) In this graph, the vertical line (x= - 1) would pass through two points . This means the graph fails the vertical line test , so it is not a function . The Vertical Line Test The vertical line test helps determine whether a graph represents a function. Rule: If a vertical line crosses the graph more than once , the relation is not a function . If every vertical line touches the graph at most once , the relation is a function .

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    Study Guide Example: (y = 2x + 3) This equation is a function . When the graph of (y = 2x + 3) is drawn: • Every vertical line intersects the graph only once . So, it passes the vertical line test . Key Ideas to Remember • A relation is a set of ordered pairs. • The domain is the set of all x - values . • The range is the set of all y - values . • A function is a relation where each input has only one output . • The vertical line test helps determine whether a graph represents a function. Summary A relation connects values using ordered pairs ((x,y)). The domain is the set of all x - values, and the range is the set of all y - values. Relations can be shown using ordered pairs, mapping diagrams, graphs, or equations. A function is a special relation in which each x - value corresponds to exactly one y - value. One way to check if a graph represents a function is by using the vertical line test . If any vertical line intersects the graph more than once, the graph does not represent a function. 2. Quiz: Basic De fi ni Ɵ ons 1. Question A relation contains the points ( - 10, 5), ( - 5, 1), (0, − 3). What is the range of that relation?

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    Study Guide Answer Choices • ( − 3, 0, 5 ) • ( − 10, − 5, 0 ) • ( − 3, 1, 5 ) Correct Answer ( − 3, 1, 5 ) Why This Is Correct The range of a relation is the set of all y - values . From the ordered pairs: ( - 10, 5 ) ( - 5, 1 ) (0, − 3 ) The y - values are: 5, 1, − 3 So the range is: { − 3, 1, 5} 2. Question A relation contains the points {(3, 2), ( − 2, 4), (5, − 3)}. What is the domain of that relation? Answer Choices • ( − 2, 3, 5 ) • ( − 3, 2, 4 ) • ( − 3, − 2, 2, 3, 4, 5 )

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