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Chapter 3: Problem 20

Find the domain and range of each relation. $$\\{(-5,-2),(0,3),(3,9),(5,12)\\}$$

Short Answer

Expert verified
The domain of the relation is \(-5, 0, 3, 5\), and the range is \(-2, 3, 9, 12\).

Step by step solution

01

Identify the domain

To find the domain, list all of the x-coordinates (the first numbers) in each ordered pair. In this relation we have -5, 0, 3, and 5 as our x-coordinates, therefore, the domain is \(-5, 0, 3, 5\).
02

Identify the range

To find the range, list all of the y-coordinates (the second numbers) in each ordered pair. In this relation we have -2, 3, 9, and 12 as our y-coordinates, therefore, the range is \(-2, 3, 9, 12\).

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Key Concepts

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

Relations in Mathematics
Relations in mathematics are a way to associate elements from one set with elements from another set. Think of it as a rule that connects each element in the first set to one or more elements in the second set.

In simple terms, a relation is just a collection of ordered pairs, where each ordered pair consists of elements from the two sets.

For instance, in the context of the exercise, the relation is expressed through the set \( \{(-5,-2),(0,3),(3,9),(5,12)\} \).

Each ordered pair connects a value from the first set (domain) with a value from the second set (range). This connection allows us to see how values relate to each other in mathematics.
Ordered Pairs
Ordered pairs are fundamental in expressing mathematical relations. An ordered pair is written in the form \((x, y)\), where the order matters.

In every ordered pair, the first element is known as the x-coordinate, and the second as the y-coordinate.

Take the ordered pair \((3, 9)\) from our list. Here, \(3\) is the x-coordinate, and \(9\) is the y-coordinate.
  • The order is essential because changing the order would change the pair meaning entirely.
  • For example, \((3, 9)\) is different from \((9, 3)\).
By structuring relations as ordered pairs, we set a clear and consistent method to visualize mathematical connections.
X-Coordinates
The x-coordinate of an ordered pair represents its horizontal position in a graph or a function. It's the first number you see in an ordered pair, detailing the input of the relationship or function.

In the set \(\{(-5,-2),(0,3),(3,9),(5,12)\}\), the x-coordinates are \(-5, 0, 3, 5\).
  • The x-coordinates create the basis of the domain of the relation.
  • They illustrate all possible independent values for the relation.
By listing these values, you define the domain, which tells us all possible x-values in our relation.
Y-Coordinates
Y-coordinates tell us the vertical position of a point on a graph, acting as the output of a function or relation. In ordered pairs, they are the second set of numbers.

From our relation \(\{(-5,-2),(0,3),(3,9),(5,12)\}\), the y-coordinates are \(-2, 3, 9, 12\).
  • The y-coordinates define what we call the range of the relation.
  • These values show us all potential outputs that can come from our input (x-coordinates).
Understanding y-coordinates is crucial in comprehending how changes in the input affect the output.

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Most popular questions from this chapter

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