Problem 28 - 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. $$\\{(-10,5),(-20,5),(-30,5)\\}$$

Step-by-Step Solution

Answer

Domain: \{-10, -20, -30\}; Range: \{5\}; It is a function.

Step 1: Identify the Relation Set Elements

The given relation is a set of ordered pairs: \((-10, 5), (-20, 5), (-30, 5)\). Each ordered pair represents the relation \(x\) to \(y\).

Step 2: Determine the Domain

The domain of a relation is the set of all first elements (\(x\)-values) from the ordered pairs. Here, the \(x\)-values are \(-10, -20,\) and \(-30\). Therefore, the domain is \(D = \{-10, -20, -30\}\).

Step 3: Determine the Range

The range of a relation is the set of all second elements (\(y\)-values) from the ordered pairs. Here, the only \(y\)-value is consistently 5. Therefore, the range is \(R = \{5\}\).

Step 4: Determine if the Relation is a Function

A relation is a function if each \(x\)-value has exactly one associated \(y\)-value. In the given relation, each \(x\)-value corresponds to exactly one \(y\)-value (5). Therefore, the relation is a function.

Key Concepts

Domain and Range

When working with functions and relations, understanding the domain and range is essential. **Domain** refers to all possible input values, often represented by the variable \(x\). For any given relation, these are the first elements in each ordered pair. In our example, the ordered pairs are \((-10, 5), (-20, 5),\) and \((-30, 5)\). Thus, the domain is \(D = \{-10, -20, -30\}\).

The **range**, on the other hand, is all the possible output values, typically represented by \(y\). These are the second elements in each ordered pair. In the given example, every ordered pair has the same second element, which is 5. This makes the range \(R = \{5\}\).
Understanding domain and range is crucial, as it helps to determine the limits and possibilities of a function or relation in modeling real-world situations.

Relations

In mathematics, a **relation** is a set of ordered pairs. It defines how the elements of two sets, usually called the domain and range, are related. Each ordered pair consists of an element from the domain and an element from the range.

Relations can be represented in various forms, such as tables, graphs, or sets of ordered pairs, like in our example. When you look at a set of ordered pairs \(\{(-10, 5), (-20, 5), (-30, 5)\}\), you see how each \(x\) value in the domain corresponds to a \(y\) value in the range.

It's important to note that relations don't always have to be functions. For a relation to be a function, each \(x\) value must map to exactly one \(y\) value. In this exercise, because each \(x\) value maps to a single \(y\) value of 5, the relation is indeed a function.

Ordered Pairs

**Ordered pairs** are fundamental in describing relations and functions. An ordered pair consists of two elements encased in parentheses, usually written as \((x, y)\). In our exercises, examples of ordered pairs are \((-10, 5)\), \((-20, 5)\), and \((-30, 5)\).

The first element in an ordered pair is the input or \(x\)-value, which belongs to the domain. The second element is the output or \(y\)-value, belonging to the range. Ordered pairs show the specific values which have a defined relationship within the given set.

They can be used to plot points on a graph and visualize the relationship between inputs and outputs. By looking at ordered pairs, we can easily observe if there is any pattern or specific rule that governs the relation. Understanding how to read and interpret ordered pairs aids significantly in analyzing functions, relations, and their respective domains and ranges.