Problem 34 - Prealgebra

Question

Determine the domain and range of the relation \(\\{(0,9),(4,8),(2,3)\) $(6,1)\\} .

Step-by-Step Solution

Answer

Domain: \(\{0, 4, 2, 6\}\); Range: \(\{9, 8, 3, 1\}\).

Step 1: Identify the Domain

The domain of a relation consists of all the first elements (x-values) in the set of ordered pairs. Examine the relation \(\{(0,9),(4,8),(2,3),(6,1)\}\). The first elements in these pairs are: \(0, 4, 2, \) and \(6\). So, the domain is the set of these numbers.

Step 2: Write the Domain

From the first step, we identified the numbers \(0, 4, 2, \) and \(6\) as the elements of our domain. We can write the domain as: \(\{0, 4, 2, 6\}\).

Step 3: Identify the Range

The range of a relation consists of all the second elements (y-values) in the set of ordered pairs. For the same relation \(\{(0,9),(4,8),(2,3),(6,1)\}\), the second elements are: \(9, 8, 3,\) and \(1\). Thus, the range is the set of these numbers.

Step 4: Write the Range

From the previous step, we found the numbers \(9, 8, 3, \) and \(1\) as our range elements. The range is: \(\{9, 8, 3, 1\}\).

Key Concepts

Domain and Range

When discussing relations and functions in math, two vital concepts are the domain and the range. The **domain** of a relation refers to all the possible input values or the first elements in each ordered pair. Essentially, it represents the **x-values** in a set of ordered pairs. For instance, in the relation \(\{(0,9),(4,8),(2,3),(6,1)\}\), the domain would be \(\{0, 4, 2, 6\}\). These values are what you can plug into a relation or function as inputs.

On the flip side, the **range** is all about the outputs. It consists of the second elements or the **y-values** in each ordered pair. These are the results you get after processing the domain through the function or relation. For the same set of ordered pairs, the range is \(\{9, 8, 3, 1\}\).

• Domain - all possible x-values or inputs of a relation
• Range - all possible y-values or outputs of a relation

Understanding domain and range is crucial when evaluating functions, as they set the groundwork for determining which inputs "work" within the given relation. By clearly identifying these two sets, you gain insights into the behavior and limitations of a function or relation.

Ordered Pairs

Ordered pairs are a fundamental concept in mathematics used to express the relationship between two quantities. In simple terms, an **ordered pair** is a set of two numbers: the first number is known as the x-value and represents the input, while the second number is the y-value, representing the output.

For example, in the ordered pair \((4, 8)\), "4" is the x-value and "8" is the y-value. The order of the numbers is important and fixed because it determines their role *(input or output)* in the relation. Hence, \((4, 8)\) is different from \((8, 4)\) because in the former, 4 serves as the input, whereas in the latter, it acts as the output.

• **X-value** - the first element representing the input
• **Y-value** - the second element representing the output

Ordered pairs are extensively used in various mathematical contexts such as graph plotting, where each pair indicates a position on the graph, and in relations or functions to depict how elements are paired. Understanding how they work is key to unlocking these applications.

Sets in Mathematics

In mathematics, a **set** is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in math, representing groups of related items. These items, referred to as elements or members, can include numbers, letters, or even other sets.

Sets are often used to define domains and ranges, or to group together elements when discussing ordered pairs. For instance, when identifying the domain \(\{0, 4, 2, 6\}\) and range \(\{9, 8, 3, 1\}\) in a relation, you are essentially creating sets of input and output values respectively.

• Elements - individual items in a set
• Notation - sets are usually written with curly braces, like \(\{a, b, c\}\)

The concept of sets helps to organize and manage groups of numbers or objects, creating a foundation for further mathematical exploration like functions and relations. Sets allow mathematicians and students alike to systematically tackle problems involving groups of data, providing clarity and structure in complex scenarios.