Problem 29 - 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. $$\\{(4,1),(3,-5),(-2,3),(3,7)\\}$$

Step-by-Step Solution

Answer

Domain: \(\{4, 3, -2\}\); Range: \(\{1, -5, 3, 7\}\); Not a function.

Step 1: Identify the Domain

The domain of a relation is the set of all first elements (x-values) from each pair. Locate these values in the relation: \(\{(4,1),(3,-5),(-2,3),(3,7)\}\). Thus, the domain \(D\) is \(\{4, 3, -2\}\). Important Note: \(3\) appears twice but each is counted once in the domain.

Step 2: Identify the Range

The range is the set of all second elements (y-values) from each pair. From the set \(\{(4,1),(3,-5),(-2,3),(3,7)\}\), extract these values. Thus, the range \(R\) is \(\{1, -5, 3, 7\}\).

Step 3: Determine if the Relation is a Function

A relation is a function if every x-value has only one unique corresponding y-value. Check the pairs: \(3\) is paired with \(-5\) and \(7\), which means an x-value corresponds to two different y-values. Therefore, the relation is not a function.

Key Concepts

Domain

The domain of a relation is an important concept in understanding functions. It refers to the set of all possible input values (commonly referred to as x-values) for which the relation is defined. In simple terms, it includes every first element from each pair in a set of ordered pairs. For example, consider the set of relations \( \{(4,1),(3,-5),(-2,3),(3,7)\} \). To determine the domain, we pick the x-values from each pair: 4, 3, and -2. Here is a critical point to remember: if an x-value repeats, such as the value 3 in this example, it is still only listed once in the domain. Hence, the domain \( D \) of this relation is

• {4, 3, -2}

This understanding helps when graphing or calculating functions, as only these values are valid for x in this particular context.

Range

Ranges are closely connected to domains and are a fundamental aspect of relations and functions. The range of a relation includes all possible output values (y-values) that result from the inputs in the domain. When you are given a set of ordered pairs, like \( \{(4,1),(3,-5),(-2,3),(3,7)\} \), to find the range, you identify the second element from each pair. In this example, the y-values are 1, -5, 3, and 7. These values collectively form the range \( R \). It's important to ensure that each y-value is counted only once, even if it appears multiple times in different pairs. Thus, the range is

• {1, -5, 3, 7}

Understanding the range allows you to determine the output values that are possible for a given relation or function, and it is crucial when analyzing the behavior and limitations of mathematical problems.

Relations

Relations form the foundation of functions in mathematics. They signify a connection between a set of inputs and corresponding outputs. When examining a relation represented as ordered pairs like \( \{(4,1),(3,-5),(-2,3),(3,7)\} \), we analyze how each x-value (input) corresponds to a y-value (output). A relation is classified as a function if every x-value is linked to exactly one unique y-value. However, in this relation, the x-value 3 has two different corresponding y-values: -5 and 7. This makes it a relation that is not a function.

Understanding the distinction between a relation and a function is crucial in mathematics, as functions have a definitive rule: one unique output for each input. Recognizing whether a relation meets this criterion is fundamental for further studies in calculus, algebra, and other mathematical disciplines. Knowing these basics can also help in the practical application of mathematical concepts in real-world scenarios.