Problem 30 - 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. $$\\{(0,5),(1,3),(0,-4)\\}$$

Step-by-Step Solution

Answer

Domain: \(\{0, 1\}\); Range: \(\{5, 3, -4\}\); Not a function.

Step 1: Identify the Domain

The domain of a relation consists of all the first elements of the ordered pairs. For the set \((0,5),(1,3),(0,-4)\), we list the first elements: \(0\) and \(1\). So, the domain is \(D = \{0, 1\}\).

Step 2: Identify the Range

The range of a relation consists of all the second elements of the ordered pairs. For the set \((0,5),(1,3),(0,-4)\), the list of second elements is \(5, 3,\) and \(-4\). Thus, the range is \(R = \{5, 3, -4\}\).

Step 3: Determine if the Relation is a Function

A relation is a function if and only if each element in the domain is associated with exactly one element in the range. In this relation, the element \(0\) in the domain is associated with two different elements in the range, \(5\) and \(-4\). Therefore, this relation is not a function because it maps a single input to multiple outputs.

Key Concepts

Relation

A relation is essentially a collection of ordered pairs. Every ordered pair consists of two elements—essentially like coordinates that solve an equation or are given as part of a dataset. In terms of ordered pairs: the first element is often termed as "input," and the second as "output." You can think of a relation as a consistent way of pairing inputs with outputs. For example, in the exercise, the relation includes the pairs

• (0, 5)
• (1, 3)
• (0, -4)

You can notice that some inputs (like 0 in this case) are associated with more than one output. This broad definition of relations allows for flexibility, meaning any set of ordered pairs qualifies as a relation, even if inputs are used more than once.

Function

Moving onto functions, these are special kinds of relations. They have a specific rule: each input can be linked to only one output. To determine if a relation is a function, we must check that every first element (or input) corresponds to exactly one second element (or output). In the exercise, while organizing the pairs

• (0, 5)
• (1, 3)
• (0, -4)

You can see that the input 0 is associated with both 5 and -4. Since one input results in multiple outputs, this particular relation is **not** a function. Functions simplify relations by creating consistency, ensuring every input has a single, predictable output. This consistency is what differentiates functions from other general relations.

Ordered Pairs

Ordered pairs are the foundation of defining relations and functions. In simpler terms, an ordered pair consists of two elements written in a specific sequence, denoted as (x, y). The first element (x) is the input, while the second element (y) is the output. These pairs help to depict mathematical relationships or even chart paths on a graph.

In the context of the exercise provided, the ordered pairs were given as

• (0, 5)
• (1, 3)
• (0, -4)

Every relation or function is essentially a collection of these ordered pairs. Analyzing the collection helps us identify the domain and range, just as the exercise demonstrated.

• The domain is represented by all the first elements: 0 and 1.
• The range includes all the second elements: 5, 3, and -4.

Because ordered pairs keep track of which inputs and outputs belong together, they are fundamental in understanding and organizing relations and functions.