Problem 20 - Elementary and Intermediate Algebra

Question

Find the domain and range of each relation. See Example 1. $$ \\{(15,-3),(0,0),(4,6),(-3,-8)\\} $$

Step-by-Step Solution

Answer

Domain: \( \{15, 0, 4, -3\} \), Range: \( \{-3, 0, 6, -8\} \).

Step 1: Identify the Domain

The domain of a relation is the set of all possible input values (x-values) of the ordered pairs. For the relation \( \{(15,-3),(0,0),(4,6),(-3,-8)\} \), list all the first components (x-values): \( 15, 0, 4, -3 \). Thus, the domain is \( \{15, 0, 4, -3\} \).

Step 2: Identify the Range

The range of a relation is the set of all possible output values (y-values) of the ordered pairs. For the relation \( \{(15,-3),(0,0),(4,6),(-3,-8)\} \), list all the second components (y-values): \( -3, 0, 6, -8 \). Thus, the range is \( \{-3, 0, 6, -8\} \).

Key Concepts

Understanding Domain

When you encounter a set of ordered pairs in algebra, the domain is an important concept to grasp. It represents the collection of all possible input values. These are the first numbers in each ordered pair, which are commonly seen as the "x-values."

• For example, in the context of algebra, if you have pairs like (15, -3), (0, 0), (4, 6), and (-3, -8), your main task is to identify the x-values.
• These values essentially tell you the boundaries or limitations of the "input" side of the pairs.

To determine the domain, list each unique x-value from the pairs:
- In our example: You have 15, 0, 4, and -3, clearly pointing out the possible x-values.
As a friendly note, repetition of values is common in larger or complex sets, but the domain remains just the unique entries. Ensuring clarity on domain makes understanding mathematical relations more approachable.

Exploring Range

Alongside the domain in ordered pairs, the range is just as crucial. It encompasses all possible output values or the set of y-values in your dataset. These y-values occur as the second component in the pairs.

• Think of it as the results or outcomes that the x-values yield.
• For a better grasp, if a set includes (15, -3), (0, 0), (4, 6), and (-3, -8), you'll zero in on -3, 0, 6, and -8, which are the respective y-values.

There you have the basics of range: it signifies the potential outcomes resulting from your x-inputs.
The key part that sometimes slips past students is ensuring no y-value is left behind or duplicated in lists. The range clarifies what outputs are feasible when input values from the domain are applied in functions or equations.

Demystifying Ordered Pairs

In mathematics, ordered pairs are foundational constructs that express relationships between two elements. Usually denoted as (x, y), the first value signifies the domain, while the second indicates the range.

• Every ordered pair provides a snapshot of a relation where an input (x) corresponds to an output (y).
• This dual structure helps in visualizing points on graphs, supporting the understanding of broader mathematical functions.

With small sets like \((15, -3), (0, 0), (4, 6), (-3, -8)\), each pair establishes a connection through the pairing of numerical values.
Understanding this concept aids in untangling various algebraic processes and enhancing your ability to solve equations or map out functions. Ordered pairs lay the groundwork, aiding both basic and advanced algebraic explorations.