Problem 28 - A Graphical Approach to College Algebra

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. $$\\{(-10,5),(-20,5),(-30,5)\\}$$

Step-by-Step Solution

Answer

Domain: \(-10, -20, -30\); Range: \(5\); Yes, it is a function.

Step 1: Identify the Domain

The domain of a relation is the set of all possible input values (x-values). For the given set of ordered pairs \{(-10,5),(-20,5),(-30,5)\}, the domain is the set of all first elements: \(-10, -20, -30\). Thus, the domain is \(D = \{-10, -20, -30\}\).

Step 2: Identify the Range

The range of a relation is the set of all possible output values (y-values). In the given set of ordered pairs \{(-10,5),(-20,5),(-30,5)\}, the range is the set of all second elements: all y-values are 5. Thus, the range is \(R = \{5\}\).

Step 3: Determine If the Relation is a Function

A relation is a function if every input value (x-value) maps to exactly one output value (y-value). Here, each x-value \(-10, -20, -30\) is paired with the same y-value of 5. Since each input has only one output, the relation is indeed a function.

Key Concepts

Domain

When we talk about the domain in mathematics, especially in the context of relations and functions, we mean all the possible input values or x-values that a relation can accept. These are usually the first numbers in the ordered pairs given within a set. In our example \{(-10, 5), (-20, 5), (-30, 5)\}, the domain includes -10, -20, and -30 because these are the x-values that appear in the pairs.
Identifying the domain is crucial because it tells us the breadth of the values we are working with. It essentially sets the boundaries on what x-values make sense in the context of the relation or function we are analyzing.

• Firstly, look at each ordered pair.
• List all unique x-values from these pairs.

In our example, the domain is \(D = \{-10, -20, -30\}\). It's important to note that there are no restrictions unless stated otherwise, but sometimes context might limit the domain further, like in real-world scenarios where negative values might not make sense.

Range

The range refers to the set of possible output values, or y-values, for a relation. It is the second component in each ordered pair. Understanding the range is as vital as the domain since it gives us a clear picture of the outputs a relation or function can yield.
For the set of pairs \{(-10, 5), (-20, 5), (-30, 5)\}, all y-values are 5. Thus, it indicates that the range is constant in this relation and is simply \(R = \{5\}\).

• Examine each ordered pair and extract the y-values.
• List all the unique output values you find.

Knowing the range helps in understanding the possible outcomes from a function or relation and in predicting how changing input values might affect output. In situations where the range is constant, like in our example, it often simplifies analysis, indicating a particular behavior of the relation.

Function Definition

Determining if a relation is a function is essential in many mathematical contexts. In simple terms, a relation is considered a function if each input (x-value) relates to exactly one output (y-value). The defining feature of a function is that no x-value leads to more than one y-value. This is often referred to as passing the "vertical line test" on a graph.
In our example set \{(-10, 5), (-20, 5), (-30, 5)\}, each distinct x-value is paired with the same y-value (5). This satisfies the definition of a function, where each x maps to only one y-value.

• Check all pairs: ensure each x appears once with a single y.
• No x should have multiple y-values.

Our relation is a function because each input value \[\{-10, -20, -30\}\] associates with one consistent output, 5. Understanding when a relation is a function helps predict the behavior of mathematical models and solve equations reliably.