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Chapter 2: Problem 3

Decide whether each relation defines a function. $$\\{(2,4),(0,2),(2,6)\\}$$

Short Answer

Expert verified
No, the relation does not define a function.

Step by step solution

01

Understand the Definition of a Function

A relation is a function if each input (x-value) is associated with exactly one output (y-value). This means that no x-value should be paired with more than one distinct y-value.
02

Identify the Pairs in the Given Relation

The given relation is: \(\{(2,4),(0,2),(2,6)\}\). The pairs are (2, 4), (0, 2), and (2, 6).
03

Check for Duplicate Inputs with Different Outputs

Examine each pair to see if there are any duplicate x-values with different y-values. In this relation, the x-value 2 appears in both (2, 4) and (2, 6) with different y-values (4 and 6).
04

Determine if the Relation is a Function

Since the x-value 2 is paired with two different y-values (4 and 6), the relation does not satisfy the criteria of a function. Therefore, this relation is not a function.

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Key Concepts

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

relation
In mathematics, a relation is a set of ordered pairs. An ordered pair consists of two components: the first component is known as the input and the second component is called the output. These components are usually referred to as \(x\)-value and \(y\)-value respectively. For example, in the relation \(\{(2,4),(0,2),(2,6)\}\), each pair \((x,y)\) shows how the values are related to each other.

Relations can be visualized using a graph, a table, or even a mapping diagram. They provide a way to show connections between elements of different sets. However, not all relations qualify as functions.
x-value
The \(x\)-value, also known as the input, is the first component in an ordered pair in a relation. It represents the independent variable that you can control or select. In the relation \(\{(2,4),(0,2),(2,6)\}\), the \(x\)-values are 2, 0, and 2.

When dealing with functions, each \(x\)-value should map to only one specific \(y\)-value. This uniqueness is crucial for distinguishing functions from general relations. Checking for repeated \(x\)-values with different \(y\)-values helps us identify if a relation is a function.
y-value
The \(y\)-value, or the output, is the second component in an ordered pair. This value is dependent on the \(x\)-value. In our example, the \(y\)-values are 4, 2, and 6.

For a relation to be a function, each \(x\)-value must be associated with exactly one \(y\)-value. If an \(x\)-value corresponds to more than one \(y\)-value, the relation cannot qualify as a function. This is demonstrated in the relation \(\{(2,4),(0,2),(2,6)\}\) where the \(x\)-value 2 is paired with both 4 and 6, violating the function criteria.
function criteria
To determine whether a relation is a function, we use the function criteria: every \(x\)-value should map to exactly one \(y\)-value. These criteria ensure the uniqueness of the function. If any \(x\)-value in the given relation pairs with more than one \(y\)-value, the relation is not a function.

Let's examine the relation \(\{(2,4),(0,2),(2,6)\}\). Here, the \(x\)-value 2 links to both 4 and 6. Because of this, the relation fails to meet the criteria of a function. Therefore, we can confidently say that the relation is not a function. Always remember that a function must have a unique output for each input.

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