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Chapter 3: Problem 1

Find the domain and the range of each relation. $$\\{(2,4),(0,0),(-7,10),(10,-7)\\}$$

Short Answer

Expert verified
The domain is \(\{2, 0, -7, 10\}\) and the range is \(\{4, 0, 10, -7\}\).

Step by step solution

01

Understand the Problem

We are given a set of ordered pairs: \((2,4), (0,0), (-7,10), (10,-7)\). Our task is to determine the domain and the range of this set of ordered pairs.
02

Identify the Domain

The domain of a relation is the set of all possible first elements (or x-values) from the ordered pairs. From \((2,4), (0,0), (-7,10), (10,-7)\), the first elements are \(2, 0, -7, 10\). Thus, the domain is \(\{2, 0, -7, 10\}\).
03

Identify the Range

The range of a relation is the set of all possible second elements (or y-values) from the ordered pairs. From \((2,4), (0,0), (-7,10), (10,-7)\), the second elements are \(4, 0, 10, -7\). Thus, the range is \(\{4, 0, 10, -7\}\).

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Key Concepts

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

Relations
In mathematics, a relation is a fundamental concept that connects two sets of information. Think of it like a pairing between elements of two sets, often represented as ordered pairs. For example, the ordered pairs given as the relation \(\{(2,4),(0,0),(-7,10),(10,-7)\}\) connect the elements of one set to elements of another. This means for each ordered pair \((x,y)\), the number \(x\) from the first set corresponds to the number \(y\) from the second set. Relations are useful for depicting real-world connections, such as assigning each person a phone number. Each individual uniquely corresponds to a number, forming a clear relationship between the two sets.
Ordered Pairs
Ordered pairs are a primary way to show relationships between two sets of numbers. Each ordered pair, like \((2,4)\), has a distinct order: the first element is the "x-value," and the second element is the "y-value." The concept of order is crucial because it shows direction: \((2,4)\) is not the same as \((4,2)\). This distinctiveness is important in defining the domain and range.
  • The first element \(x\) is the "input" or "starting point."
  • The second element \(y\) is the "output" or "result."
Ordered pairs help in visualizing mathematical functions and relations through Cartesian coordinates, where each pair pinpoints a location on a graph.
Set Notation
Set notation is a compact and precise way of describing collections of objects, such as numbers or ordered pairs. It's characterized by the use of curly braces \(\{ \} \) and commas, which list the elements within a set. In the exercise, the domain and range of the relation are expressed as sets. For instance:
  • The domain \(\{2, 0, -7, 10\}\) indicates all the x-coordinates from each ordered pair.
  • The range \(\{4, 0, 10, -7\}\) lists all the y-coordinates.
Set notation makes it easier to understand and communicate mathematical ideas because it clearly shows elements belonging to a set without ambiguity. This notation helps students and mathematicians alike to identify and classify multiple elements quickly.

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Most popular questions from this chapter

For Exercises 87 through \(91,\) fill in each blank with "0, "positive," or "negative." For Exercises 92 and \(93,\) fill in each blank with "x" \(o r^{\text {tr }} y^{\prime \prime}\) (__,__) quadrant II

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