Problem 22 - Elementary and Intermediate Algebra

Question

Find the domain and range of each relation. See Example 1. $$ \\{(1,-12),(-6,8),(5,8),(0,0),(1,4)\\} $$

Step-by-Step Solution

Answer

Domain: \( \{1, -6, 5, 0\} \); Range: \( \{-12, 8, 0, 4\} \).

Step 1: Understand the Relation

A relation is a set of ordered pairs. Here, the relation is given as \( \{(1,-12),(-6,8),(5,8),(0,0),(1,4)\} \). Each pair consists of an 'input' (first element) and an 'output' (second element).

Step 2: Identify the Domain

The domain of the relation is the set of all 'inputs' or first elements from each ordered pair. From the given pairs: \(1, -6, 5, 0, 1\). Since the number 1 repeats, we list each unique input only once. Thus, the domain is \( \{1, -6, 5, 0\} \).

Step 3: Identify the Range

The range of the relation is the set of all 'outputs' or second elements from each ordered pair. From the given pairs: \(-12, 8, 8, 0, 4\). We list each unique output only once. Thus, the range is \( \{-12, 8, 0, 4\} \).

Key Concepts

Relations

In mathematics, a relation is a set of ordered pairs. These can be thought of as links between two sets of information. For example, in the set \((1,-12),(-6,8),(5,8),(0,0),(1,4)\), each ordered pair represents a specific relation between two values. One value, known as the 'input', is mapped to another value, the 'output'.
Understanding relations is crucial because they form the basis for more advanced concepts like functions. Not all relations are functions, but every function is a relation with a special property: each input is related to exactly one output.
When you're given a set of ordered pairs and asked about the domain and range, you're really exploring the relationship between these inputs and outputs.

Ordered Pairs

Ordered pairs are foundational to the study of relations. An ordered pair, like \((1, -12)\), shows a direct relationship where the first element is the input, and the second is the output.
It's important to remember the sequence in an ordered pair matters: \((a, b)\) is different from \((b, a)\). In our example, the input \(1\) corresponds to the output \(-12\), meaning that when the input is \(1\), the result or output is \(-12\).
Ordered pairs are represented in the Cartesian coordinate system as points on a graph, where the x-axis represents the input and the y-axis represents the output. This visual representation helps us quickly see how each input corresponds to an output and analyze the relationship between the sets.

Input and Output

In the context of relations represented by ordered pairs, the term 'input' refers to the first element of each pair, while 'output' refers to the second element. Let's look deeper into what these two terms imply.

• Input: Also known as the domain of the relation, these are the independent variables or set of initial values. In our example, the inputs are \(1, -6, 5, 0, 1\). The domain, therefore, includes each distinct input: \(\{1, -6, 5, 0\}\).
• Output: These are dependent variables that result from the inputs, known collectively as the range. For the ordered pairs given, the outputs are \(-12, 8, 8, 0, 4\), which results in the range \(\{-12, 8, 0, 4\}\).

Recognizing the input and output values allows us to map how changes in one set (input) affect the other set (output), thereby defining the concept of the relation itself.