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Chapter 2: Problem 2

Determine the domain and range and state whether the relation is a function or not. $$\\{(2,0),(4,3),(6,6),(8,6),(10,9)\\}$$

Short Answer

Expert verified
Domain: \(-3, -2, 1, 2\); Range: \(0, 1, 3, 7, 5\); Not a function.

Step by step solution

01

Identify the Domain

The domain is the set of all first elements (x-values) from the given set of ordered pairs. From the set \((-3,0), (-2,1), (1,3), (2,7), (2,5)\), the domain is \(-3, -2, 1, 2\).
02

Identify the Range

The range is the set of all second elements (y-values) from the given set of ordered pairs. From the set \((-3,0), (-2,1), (1,3), (2,7), (2,5)\), the range is \(0, 1, 3, 7, 5\).
03

Determine if It's a Function

A function has only one unique y-value for each x-value. Identify any repeating x-values with different y-values in the set. The x-value '2' has y-values '7' and '5,' so this set is not a function because it repeats x-values with different outputs.

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Key Concepts

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

Functions
In mathematics, a function is a special type of relationship. It maps each input to exactly one output. This means for each x-value, there is only one y-value. If any x-value corresponds to more than one y-value in a set of ordered pairs, then it isn't a function.

To determine if a relation is a function, check the x-values:
  • If no x-value repeats, it is a function.
  • If an x-value appears multiple times but always corresponds to the same y-value, it is still a function.
  • If an x-value appears with different y-values, it is not a function.
In the example of \((-3,0), (-2,1), (1,3), (2,7), (2,5)\), the x-value '2' maps to two different y-values, '7' and '5'. Because of this inconsistency, this set is not a function.
Relations
A relation is any set of ordered pairs. It describes how elements from one set (usually x-values) are related to elements in another set (usually y-values). All functions are relations, but not all relations are functions.

In this sense, relations can show:
  • Mappings where each x-value might have multiple y-values.
  • Establishing connections or associations between items.
  • Relationships where y-values might be paired with multiple x-values.
The given set \((-3,0), (-2,1), (1,3), (2,7), (2,5)\) is a relation because it is a collection of ordered pairs where each x-value has a corresponding y-value.
Ordered Pairs
An ordered pair is a pair of elements organized as \((x, y)\). The first element is the x-value, and the second element is the y-value. Ordered pairs are a simple way to represent points or data in two-dimensional space.

Some things to remember about ordered pairs are:
  • The order matters: \((3, 2)\) is distinct from \((2, 3)\).
  • Used primarily in coordinate geometry to plot points on a graph.
  • Each pair in a relation or function consists of one x-value and one y-value.
In the problem, \((-3,0), (-2,1), (1,3), (2,7), (2,5)\) are examples of ordered pairs, with each pair representing a point or entry in the relation.
X-value
The x-value in an ordered pair \((x, y)\) reflects the horizontal position on a graph. In the context of relations and functions, x-values indicate the input or domain of a given set.

Key points about x-values:
  • Also known as the "input" or "independent variable".
  • Forms the first element of an ordered pair \((x, y)\).
  • In a function, each x-value must map to exactly one y-value.
In the set \((-3,0), (-2,1), (1,3), (2,7), (2,5)\), the x-values are \(-3, -2, 1, 2\).
Y-value
The y-value in an ordered pair \((x, y)\) indicates the vertical position on a graph. It is often referred to as the output or range in the context of relations and functions.

Important aspects of y-values include:
  • Known as the "output" or "dependent variable".
  • Represented as the second element in an ordered pair \((x, y)\).
  • The y-value can be the same for different x-values in a relation.
Based on the given set \((-3,0), (-2,1), (1,3), (2,7), (2,5)\), the y-values are \(0, 1, 3, 7, 5\).

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