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Chapter 2: Problem 22

Find the domain and range of each relation. \(\\{(1,-12),(-6,8),(5,8),(0,0),(1,4)\\}\)

Short Answer

Expert verified
Domain: \{-6, 0, 1, 5\}; Range: \{-12, 0, 4, 8\}.

Step by step solution

01

- Identify All First Elements

The first elements in each ordered pair represent the domain of the relation. List them out: 1, -6, 5, 0, and 1.
02

- List Unique Domain Elements

Identify unique elements from the first elements list. The domain of the relation is \(\{-6, 0, 1, 5\}\), as we list unique elements from the first set.
03

- Identify All Second Elements

The second elements in each ordered pair represent the range of the relation. List them out: -12, 8, 8, 0, and 4.
04

- List Unique Range Elements

Identify unique elements from the second elements list. The range of the relation is \(\{-12, 0, 4, 8\}\), as we list unique elements from the second set.

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Key Concepts

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

Understanding Relations
A relation is any set of ordered pairs, like the ensemble you鈥檝e been provided with in your exercise. Relations in mathematics essentially connect two sets of information. In simpler terms, it pairs every element of one set with an element from another set.
  • Relations can be visualized as arrows pointing from elements in one set to elements in another.
  • For the relation \(\{(1,-12),(-6,8),(5,8),(0,0),(1,4)\}\), the connection is direct: each ordered pair connects a member of the domain with a member of the range.
Understanding how these connections work is crucial. Each link signifies some kind of relationship. This could be numbers corresponding in a dataset, functions linking input and output, or anything that has a one-to-one match or several-to-one match.
Ordered Pairs Explained
An ordered pair is a pair of numbers that, together, represent a point in a plane or a direct relationship between inputs and outputs. The notation \((a, b)\) means that \('a'\) is related to \('b'\). These are crucial in identifying domains and ranges in relations.
  • The first number in an ordered pair is often referred to as the input or domain element.
  • The second number is typically the output or range element.
For example, in the pair (1, -12), \(1\) is our input and \(-12\) is the output. Understanding ordered pairs helps us easily group these connections and visualize the relationships in a graphical format if needed. Each pair basically tells a tiny story of relationship in a larger narrative.
The Beauty of Unique Elements
Recognizing unique elements within a relation is essential in determining the domain and range. In any given set, finding unique elements means identifying all distinct numbers without repetitions.
  • In a mathematical relation, this is how we streamline the data to capture the purest form of information.
  • This process helps develop a clearer understanding of the relationships at play.
From the provided exercise, for the domain (first elements): \(\{1, -6, 5, 0, 1\}\), the uniqueness is found in \(\{-6, 0, 1, 5\}\) after removing duplicates. Similarly, for the range (second elements): \(\{-12, 8, 8, 0, 4\}\), the range is \(\{-12, 0, 4, 8\}\).
Spotting these unique elements not only simplifies the complexity of the relation but also enhances clarity in interpreting its true domain and range.

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