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Chapter 2: Problem 45

Determine the domain and range of each relation. $$\\{(x, y) | y=4\\}$$

Short Answer

Expert verified
Domain: 鈩� (all real numbers), Range: {4}.

Step by step solution

01

Understand the Relation

The given relation is described by the set of all points \((x, y)\) where the y-coordinate is always 4.
02

Identify the Domain

The domain of a relation is the set of all possible x-values. Since no restrictions are put on x, x can be any real number. Therefore, the domain is all real numbers, denoted as \(\text{Domain} = \text{鈩潁\) or \(\text{Domain} = (-\text{鈭瀩, \text{鈭瀩)\).
03

Identify the Range

The range of the relation is the set of all possible y-values. In this relation, y is always equal to 4. Thus, the range is exactly the single value 4, written as \(\text{Range} = \{4\}\).

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Key Concepts

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

relations in mathematics
A relation in mathematics is a collection of ordered pairs, usually written as \(x, y\). This represents a connection between elements of different sets. For example, the ordered pair (2, 4) signifies that the element 2 from one set is related to the element 4 from another set.
Relations can be visualized on the coordinate plane. Each pair (x, y) is a point that can be plotted to see how elements are connected. You can think of relations as mathematical rules that link inputs to outputs.
In our exercise, the relation is y = 4. This means every ordered pair has the y-coordinate 4, representing a horizontal line crossing y=4.
real numbers
Real numbers \( \text{鈩潁 \) are all the numbers you can think of on the number line. They include both rational numbers (like 1, 3/2, 0.75) and irrational numbers (like \( \text{鈭�2} \), \( \text{蟺} \)).
Real numbers are important in mathematics because they allow us to measure and compare quantities continuously. When talking about the domain and range, we often refer to real numbers for all possible x-values and y-values.
Our exercise involves determining the domain and range within the scope of real numbers.
domain of a function
The domain of a function is the set of all possible input values (x-values) that the function can accept. For the relation given in our exercise \{ (x, y) | y = 4 \}, the domain is the set of all real numbers because there are no limitations on the x-values.
We denote this as \( \text{Domain} = \text{鈩潁 = (-\text{鈭瀩, \text{鈭瀩) \). This means x can be any number, whether positive, negative, or zero.
Understanding the domain helps us know where the relation or function exists on the x-axis.
range of a function
The range of a function is the set of all possible output values (y-values) that the function can produce. Referring to our exercise with the relation \{ (x, y) | y = 4 \}, the range is straightforward. Since y is always 4, there is only one value in the range.
We express this as \( \text{Range} = \{ 4 \} \).
The range tells us how the outputs (y-values) are distributed. In this case, it simplifies to a single point on the y-axis, helping us understand its behavior on a graph.

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