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Chapter 2: Problem 46

Determine the domain and range of each relation. $$\\{(x, y) | x=5\\}$$

Short Answer

Expert verified
Domain: \{5\}, Range: \mathbb{R}\.

Step by step solution

01

Identify the Given Relation

The given relation is \{(x, y) | x=5\}. This means that for every pair \(x, y\) in the relation, the value of \(x\) is always 5.
02

Determine the Domain

The domain is the set of all possible values of \(x\) in the relation. Since \(x\) is always 5, the domain is \{5\}.
03

Determine the Range

The range is the set of all possible values of \(y\) in the relation. Since \(y\) can be any value when \(x = 5\), the range is the set of all real numbers, \mathbb{R}\.

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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 simply a set of ordered pairs \(x, y\). Each pair consists of two components: an input (x) and an output (y). A relation defines a connection between these pairs based on a certain rule or condition. For instance, in the relation \{(x, y) | x=5\}, the condition is that the x-value is always 5. This means no matter what y is, x remains constant at 5. This set-up can help us see how one element can relate to another in a consistent way. Relations are foundational because they help us understand more complex functions and mathematical concepts. Some common examples of relations include equations, inequalities, and mappings.
Domain
The domain of a relation refers to the collection of all possible input values (or x-values) that the relation can accept. In simpler terms, it is what you are allowed to plug into the relation. For the given example \(\{(x, y) | x=5\}\), since x must always be 5, there is just one possible value for x. Therefore, the domain of this relation is \[ \{ 5 \} \]. Determining the domain is crucial because it gives insight into the extent or limits of the relation. It's akin to setting the boundaries for what inputs make the relation work.
Range
The range of a relation is the set of all possible output values (or y-values) that you can get by plugging in the domain values into the relation. For the relation \{(x, y) | x=5\}, while x is fixed at 5, y can be any real number. Thus, the range in this example is the entire set of real numbers, denoted as \(\backslash mathbb{R} \). Understanding the range is essential as it tells us the spread of possible outputs or results we can achieve from the relation. In essence, it helps us see the full picture of what the relation can produce.
Real Numbers
Real numbers are a broad classification of numbers that includes both rational and irrational numbers. They encompass all the numbers you can think of that exist on the number line, including positive numbers, negative numbers, and zero. This set is denoted as \(\backslash mathbb{R} \). For the relation \{(x, y) | x=5\}, when we say the range is the set of all real numbers, we imply that y can take any value from \(\backslash mathbb{R} \). This makes the concept of real numbers very versatile and significant in understanding mathematical relations and more complex functions.

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