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A relation is given by the set of ordered pairs: R = (-2, 3), (-1, 4), (1, 7), (2, 7). a) Write down the domain and range of R. b) Define the relation represented by R. c) Represent this relation usin

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2 parts
a

{−2,−1,1,2}\{-2, -1, 1, 2\}

{3,4,7}\{3, 4, 7\}

d

function (many-to-one)\text{function (many-to-one)}

Step-by-step solution

4 steps

here are the solutions to problems 6 and 7.

:
A relation is given by the set of ordered pairs: R={(−2,3),(−1,4),(1,7),(2,7)}R = \{(-2, 3), (-1, 4), (1, 7), (2, 7)\}.

Part a· 2 steps

To write down the domain and range of R.

  1. Identify the domain.

    The domain is the set of all first elements (x-coordinates) from the ordered pairs.

    Domain={−2,−1,1,2}\text{Domain} = \boxed{\{-2, -1, 1, 2\}}

  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}\text{Range} = \boxed{\{3, 4, 7\}}

Part 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.

Part 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}\{-2, -1, 1, 2\} inside the first oval.
  • List the elements of the Range {3,4,7}\{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 −2-2 to 3
    • Arrow from −1-1 to 4
    • Arrow from 1 to 7
    • Arrow from 2 to 7

Part d

function (many-to-one)\text{function (many-to-one)}

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)\boxed{\text{function (many-to-one)}}.

:
Draw the graph of the function f(x)=3x−5f(x) = 3x - 5 for the domain 0≤x≤2,x∈R0 \le x \le 2, x \in \mathbb{R}.

  1. Find the coordinates of the endpoints of the line segment by substituting the boundary values of the domain into the function.

    For x=0x = 0:

    f(0)=3(0)−5=−5f(0) = 3(0) - 5 = -5

    This gives the point (0,−5)(0, -5).

    For x=2x = 2:

    f(2)=3(2)−5=6−5=1f(2) = 3(2) - 5 = 6 - 5 = 1

    This gives the point (2,1)(2, 1).

  2. Describe how to draw the graph.

    • Draw a Cartesian coordinate plane with clearly labeled x and y axes.
    • Plot the point (0,−5)(0, -5).
    • Plot the point (2,1)(2, 1).
    • Draw a straight line segment connecting these two points. Since the domain is 0≤x≤20 \le x \le 2 (inclusive), the line segment should include both endpoints.
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