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Chapter 3: Problem 26

Find the domain and range of each relation. Then determine whether the relation represents a function. \\{(0,-2),(1,3),(2,3),(3,7)\\}

Short Answer

Expert verified
Domain: \(\{0, 1, 2, 3\}\). Range: \(\{-2, 3, 7\}\). This relation is a function.

Step by step solution

01

Identify the Domain

The domain of a relation is the set of all possible input values, or x-values, from the ordered pairs. For the given relation \((0,-2),(1,3),(2,3),(3,7)\), identify the x-values: \(\{0, 1, 2, 3\}\).
02

Identify the Range

The range of a relation is the set of all possible output values, or y-values, from the ordered pairs. For the given relation \((0,-2),(1,3),(2,3),(3,7)\), identify the y-values: \(\{-2, 3, 7\}\).
03

Determine if the Relation is a Function

A relation is a function if every x-value corresponds to exactly one y-value. Check each x-value in the given relation: - For \(x=0\), the y-value is \(-2\).- For \(x=1\), the y-value is \(3\).- For \(x=2\), the y-value is \(3\).- For \(x=3\), the y-value is \(7\).Since each x-value has a single corresponding y-value, the relation is a function.

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Key Concepts

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

Relation
A relation is essentially a collection of ordered pairs. If you think of a pair like a couple, the first number in the pair is linked to the second number. Each ordered pair thus signifies a connection between two elements. For instance, in the ordered pair \(3, 7\), the number 3 is linked or related to the number 7.
To break this down further:
  • The first number in each pair represents an input value.
  • The second number in each pair represents an output value.
In our given example, we have the relation \((0, -2), (1, 3), (2, 3), (3, 7)\). Here, the input value 0 is associated with the output value \(-2\), the input value 1 is associated with 3, and so on. By examining these pairs, we can understand how each input is mapped to an output.
Function
A function is a special type of relation where each input value (x-value) has exactly one unique output value (y-value). In other words, no input value can be linked to more than one output value.
Consider the given relation \((0, -2), (1, 3), (2, 3), (3, 7)\). To check if this relation is a function, follow these steps:
  • Start with \x = 0\. It maps to \y = -2\.
  • Next, \x = 1\ maps to \y = 3\.
  • Then, \x = 2\ maps to \y = 3\.
  • Finally, \x = 3\ maps to \y = 7\.
Inspect each pair: since each x-value corresponds to one single y-value, this relation is indeed a function. A key takeaway is: if any x-value maps to multiple y-values, it鈥檚 not a function.
Ordered Pairs
Ordered pairs are the building blocks of relations and functions. Each pair consists of two numbers written in a specific order鈥攗sually as \(x, y\).
  • The first number in the pair is called the input value or the domain element (x-value).
  • The second number in the pair is called the output value or the range element (y-value).
In our example, \(1, 3\) is an ordered pair where 1 is the input and 3 is the output. These pairs help us understand the relationships between different values.
Input Values
Input values, also known as domain elements, are the first numbers in our ordered pairs. They are the values that we 'put into' a function or relation to receive an output. In our example, \((0, -2), (1, 3), (2, 3), (3, 7)\), the input values are \{0, 1, 2, 3\}.
  • Each input value appears only once in our list of ordered pairs.
  • Input values are crucial because they help determine the corresponding output values in a relation.
The input values provide the foundation for understanding the relationship between variables in functions and relations.
Output Values
Output values, or range elements, are the second numbers in ordered pairs. They are the results we get after applying the input values to a function or relation. For the given pairs \((0, -2), (1, 3), (2, 3), (3, 7)\), the output values are \{-2, 3, 7\}.
  • The output values can repeat, as seen with 3 in this case.
  • They showcase what each input value yields when processed by the function or relation.
Understanding output values helps identify the range and can help diagnose how a function or relation behaves with different inputs.

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Most popular questions from this chapter

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