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Chapter 10: Problem 1

Find the domain and range of the relation. $$\\{(-4,3),(2,5),(1,2),(4,-3)\\}$$

Short Answer

Expert verified
The domain of the relation is \{-4, 2, 1, 4\} and the range of the relation is \{3, 5, 2, -3\}.

Step by step solution

01

Identify Domain

The first element in each ordered pair represents the x-coordinate, which are the elements of the domain. So in the given relation \{(-4,3),(2,5),(1,2),(4,-3)\}, the x-coordinates are -4, 2, 1, and 4.
02

Identify Range

The second element in each ordered pair represents the y-coordinate, which are the elements of the range. So in the given relation \{(-4,3),(2,5),(1,2),(4,-3)\}, the y-coordinates are 3, 5, 2, and -3.
03

Write down Domain and Range

After having identified all x-coordinates and y-coordinates, they can respectively be written as the domain and the range of the relation.

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Key Concepts

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

Ordered Pairs
An ordered pair is a fundamental concept in mathematics, especially when dealing with relations and functions. Ordered pairs consist of two elements, typically written as \((x, y)\). This notation means that each pair contains two values, one representing the horizontal position (x-coordinate) and the other representing the vertical position (y-coordinate).

  • The position of the elements in the pair is crucial, as altering them from \((x, y)\) to \((y, x)\) can change their meaning entirely.
  • Ordered pairs are used to denote points on the Cartesian coordinate system, where the first number represents the position along the x-axis and the second along the y-axis.
  • In the context of relations, ordered pairs help in identifying domains and ranges.
Understanding ordered pairs is essential for identifying how elements are interconnected, especially within mathematical relations.
X-coordinate
The x-coordinate is the first number in an ordered pair. It determines the horizontal position of a point in the Cartesian plane. For example, in the ordered pair \((-4, 3)\), \(-4\) is the x-coordinate.

  • The x-coordinate indicates how far left or right a point is from the origin \((0,0)\) on the x-axis.
  • If the x-coordinate is negative, the point is positioned to the left of the origin.
  • If the x-coordinate is positive, the point lies to the right of the origin.
In the given relation \{(-4,3),(2,5),(1,2),(4,-3)\}, the x-coordinates \(-4, 2, 1,\) and \(4\) form the domain, representing all possible values for the horizontal axis.
Y-coordinate
The y-coordinate is the second number in an ordered pair, and it signifies the vertical position of a point in space. For instance, in \((-4, 3)\), \(3\) is the y-coordinate.

  • The y-coordinate shows how far a point is above or below the origin along the y-axis.
  • If the y-coordinate is positive, the point is above the origin.
  • If the y-coordinate is negative, it is below the origin.
For the given relation \{(-4,3),(2,5),(1,2),(4,-3)\}, the y-coordinates \(3, 5, 2,\) and \(-3\) make up the range. This set contains all possible values for the vertical axis, effectively mapping the response in each ordered pair.

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