Problem 26 - A Graphical Approach to Precalculus with Limits

Question

Determine the domain \(D\) and range \(R\) of each relation, and tell whether the relation is a function. Assume that a calculator graph extends indefinitely and a table includes only the points shown. $$\\{(8,0),(5,4),(9,3),(3,8)\\}$$

Step-by-Step Solution

Answer

Domain: \(\{8, 5, 9, 3\}\); Range: \(\{0, 4, 3, 8\}\); The relation is a function.

Step 1: Identify Domain

The domain of a relation is the set of all first elements (x-values) from the pairs. Here, the relation consists of the points \((8,0),(5,4),(9,3),(3,8)\). Thus, the domain, \(D\), is \(\{8, 5, 9, 3\}\).

Step 2: Identify Range

The range of a relation is the set of all second elements (y-values) from the pairs. In this case, the range, \(R\), is \(\{0, 4, 3, 8\}\) based on the given points.

Step 3: Determine if the Relation is a Function

Recall that a relation is a function if each input (x-value) is paired with exactly one output (y-value). Here, every x-value in the set \(\{8, 5, 9, 3\}\) maps to a unique y-value, meaning no x-value is repeated with a different y-value. Therefore, this relation is a function.

Key Concepts

Understanding the Domain in a Relation

The **domain** of a relation is a key foundational concept in understanding functions and relations in mathematics. Essentially, the domain refers to the complete set of possible input values (typically x-values) that a relation can accept. Think of it as the ingredients you have to make a dish—you can't make the dish without them.
The exercise provided points like \((8,0), (5,4), (9,3), (3,8)\). Here, the domain is simply the list of all the x-values from these pairs: \(\{8, 5, 9, 3\}\).

• In a graphical approach, this usually represents all the possible x-values where the graph extends along the x-axis.
• In a tabular form like our exercise, you just pick all the first values from the pairs.

Understanding the domain helps you recognize the limits of the inputs for a function or relation. It ensures that you always know the scope of the data you can work with.

Exploring the Range in a Relation

Just as important as the domain, the **range** represents all the possible output values (typically y-values) that a relation can produce. You can think of it as the outcomes or results we get from our inputs.
In our given set of points \((8,0), (5,4), (9,3), (3,8)\), the range is obtained from the y-values, which are \(\{0, 4, 3, 8\}\). This tells us the set of all the possible outputs from the relation.

• In a graph, this includes the y-values that the relation reaches or touches.
• In a table, you will just extract all the second elements in each pair, much like we did above.

Knowing the range allows you to understand what kind of values a function or relation can generate, which is crucial when analyzing its behavior or characteristics in a broader context.

Determining Relations and Functions

The concept of a relation becomes even more focused when we consider whether it is a function. A **relation** in mathematics simply describes any set of ordered pairs. However, not every relation is a function. A function is a special type of relation where each input corresponds exactly to one output.
In the exercise, the relation given is \(\{(8,0),(5,4),(9,3),(3,8)\}\). To tell if this is a function, we need to ensure that no x-value is repeated with a different y-value. Here, each x-value is paired with one unique y-value, meaning it passes the function test!

• A function can be imagined like a vending machine where pressing one button gives only one item.
• If pressing a button in the machine gives out multiple different items randomly, it is no longer functioning properly - it's just a relation, not a function.

Understanding this concept helps solidify the unique role functions play in mathematics, allowing you to model consistent outputs for given sets of inputs.