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Chapter 3: Problem 3

Find the domain and the range of each relation. $$ \\{(0,-2),(1,-2),(5,-2)\\} $$

Short Answer

Expert verified
Domain: \( \{0, 1, 5\} \); Range: \( \{-2\} \).

Step by step solution

01

Identify the Domain

The domain of a relation is the set of all possible input values (x-values) that the relation can accept. To find the domain, we need to list all the first elements from each ordered pair in the given set. The ordered pairs are (0,-2), (1,-2), and (5,-2). The x-values are 0, 1, and 5. Therefore, the domain is \( \{0, 1, 5\} \).
02

Identify the Range

The range of a relation is the set of all possible output values (y-values) that the relation can output. To find the range, we need to list all the second elements from each ordered pair in the given set. For the ordered pairs (0,-2), (1,-2), and (5,-2), the y-values are all -2. Since all the y-values are the same, the range is \( \{-2\} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Ordered Pairs
Ordered pairs are a fundamental concept in mathematics, often used to express relationships between two elements. These pairs consist of two numbers separated by a comma and enclosed in parentheses, like \(0, -2\). These numbers represent positions in a coordinate system.
  • First element (x-value): Represents the horizontal position on a graph or the input in a relation.
  • Second element (y-value): Represents the vertical position on a graph or the output in a relation.
Each ordered pair gives a precise way of expressing mathematical relationships. This clarity is crucial when analyzing domains and ranges, as each element must be considered in the context of its pair.
Exploring X-Values
The term x-values refers to the first elements in our ordered pairs. These values form the domain of a relation. In the set of ordered pairs \(0, -2\), \(1, -2\), and \(5, -2\), the x-values are 0, 1, and 5.
  • Identifying x-values: For any set of ordered pairs, looking at the first number tells us the x-value.
  • Domain: The collection of all x-values from a given set forms the domain. In our example, the domain is \(\{0, 1, 5\}\).
Understanding the x-values helps us recognize which inputs are possible for the relation and their scope within a coordinate system.
Demystifying Y-Values
Y-values are the second elements in an ordered pair, describing the outputs of a relation. These form what is known as the range. For the pairs \(0, -2\), \(1, -2\), and \(5, -2\), the y-value consistently remains -2.
  • Identifying y-values: In any ordered pair, the number following the comma represents the y-value.
  • Range: The set of all y-values forms the range of the relation. Manifested as \(\{-2\}\) in our example, because all y-values are the same.
Grasping the concept of y-values is crucial for understanding the outputs that can be produced from the inputs within a relation.

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