This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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You're on a roll — here are the solutions to problems 6 and 7. Problem 6: A relation is given by the set of ordered pairs: R = \(-2, 3), (-1, 4), (1, 7), (2, 7)\. a) To write down the domain and range of R. Step 1: Identify the domain. The domain is the set of all first elements (x-coordinates) from the ordered pairs. Domain = \-2, -1, 1, 2\ Step 2: Identify the range. The range is the set of all second elements (y-coordinates) from the ordered pairs, listed without repetition. Range = \3, 4, 7\ b) To define the relation represented by R. A relation is a set of ordered pairs that shows a correspondence between elements of two sets. Specifically, it maps elements from a domain set to elements in a range set. c) To represent this relation using an arrow diagram. To draw an arrow diagram: • Draw two ovals. Label the first oval "Domain" and the second "Range". • List the elements of the Domain \-2, -1, 1, 2\ inside the first oval. • List the elements of the Range \3, 4, 7\ inside the second oval. • Draw an arrow from each element in the Domain to its corresponding element in the Range, based on the given ordered pairs: • Arrow from -2 to 3 • Arrow from -1 to 4 • Arrow from 1 to 7 • Arrow from 2 to 7 d) To state what type of relation R is. A relation is a function if each element in the domain maps to exactly one element in the range. In this relation, each x-value has only one y-value associated with it. For example, 1 maps to 7, and 2 also maps to 7, but neither 1 nor 2 maps to any other value. Therefore, R is a function. More specifically, it is a many-to-one function because multiple domain elements (1 and 2) map to the same range element (7). The type of relation is a function (many-to-one). Problem 7: Draw the graph of the function f(x) = 3x - 5 for the domain 0 x 2, x R. Step 1: Find the coordinates of the endpoints of the line segment by substituting the boundary values of the domain into the function. For x = 0: f(0) = 3(0) - 5 = -5 This gives the point (0, -5). For x = 2: f(2) = 3(2) - 5 = 6 - 5 = 1 This gives the point (2, 1). Step 2: Describe how to draw the graph. • Draw a Cartesian coordinate plane with clearly labeled x and y axes. • Plot the point (0, -5). • Plot the point (2, 1). • Draw a straight line segment connecting these two points. Since the domain is 0 x 2 (inclusive), the line segment should include both endpoints. What's next?