Problem 9 - Algebra 2

Question

Make a mapping diagram for each relation. $$ \\{(-2,8),(-1,1),(0,0),(1,1),(2,8)\\} $$

Step-by-Step Solution

Answer

The mapping diagram will consist of a 'Domain' column with numbers -2, -1, 0, 1, 2 and a 'Range' column with numbers 0, 1, 8. Arrows will connect -2 to 8, -1 to 1, 0 to 0, 1 to 1, and 2 to 8, with some elements in the domain mapping to the same number in the range.

Step 1: Identify the Domain and Range

The first step is to identify the domain (set of all the first elements in each ordered pair) and range (set of all the second elements in each ordered pair). In this relation, the domain is \(\{-2, -1, 0, 1, 2\}\) and the range is \(\{8, 1, 0\}\).

Step 2: Set up the Mapping Diagram

Start by drawing two parallel vertical lines or columns. Label one column 'Domain' and the other 'Range'.

Step 3: Plot the Domain and Range

On the Domain column, list the numbers -2 to 2 vertically with equal spacing. On the Range column, list the numbers 0, 1, and 8, making sure to align them horizontally with the corresponding domain values.

Step 4: Draw Arrows for the Relationship

Draw arrows from each number in the Domain column to its corresponding number in the Range column based on the given pairs. Note that there will be arrows from -2 to 8, -1 to 1, 0 to 0, 1 to 1, and 2 to 8.

Step 5: Check for Mistakes

Make sure that each element of the domain is connected to the correct element of the range according to the given relation. There should be two pairs of identical mappings: \( -1 \to 1 \) and \( 1 \to 1 \) as well as two pairs of identical mappings: \( -2 \to 8 \) and \( 2 \to 8 \).

Key Concepts

Domain and Range

Understanding the domain and range of a relation is fundamental in algebra. The domain includes all the possible inputs, or the x-values, from ordered pairs. In contrast, the range comprises all possible outputs, or y-values.

When listing the domain and range from a set of ordered pairs, like the given exercise \(\{-2, -1, 0, 1, 2\}\) for domain and \(\{8, 1, 0\}\) for range, we are essentially summarizing all the unique starting points and their potential outcomes in the relation. Notably, each element in the domain corresponds to at least one element in the range. However, not every element in the range must be linked to a different domain element; some can repeat, as illustrated by the ordered pairs \(\{-2,8\}\) and \(\{2,8\}\) sharing the same range value of 8.

Ordered Pairs

An ordered pair is the basic building block of relations in algebra, written as \( (x, y) \) where \( x \) represents an element from the domain and \( y \) from the range. The order in which these pairs are written is crucial; the first element always comes from the domain, and the second from the range.

The relationship can be visualized as a link between these two sets. In the mapped relation from the example \(\{(-2,8),(-1,1),(0,0),(1,1),(2,8)\}\), each of these connections is shown by an arrow in the mapping diagram. This representation conveys how each input is related to an output in a clear, visual manner.

Relation in Math

A relation in math describes the connection between two sets of information. It is composed of ordered pairs where the first element is associated with the second. Not all relations are functions, but all functions are relations.

For clarity, a function has a critical rule: each input from the domain is associated with exactly one output from the range. The given relation in the exercise can be a function since each domain number is paired with only one range number. However, if a single domain element were linked to multiple range values, it would not meet the definition of a function.

Function Representation

A function is a special type of relation that pairs each domain value with a single unique range value. This relation can be represented in various ways, such as equations, graphs, tables, and mapping diagrams.

The mapping diagram is particularly helpful as it visually displays the domain to range connections, confirming whether a relation is a function. For example, in our exercise, drawing arrows between corresponding domain and range values on a mapping diagram quickly reveals the nature of the relationship. Remember, if any domain value pointed to more than one range value, it would not represent a function. This clarification is essential for students as they learn to distinguish between different kinds of mathematical relationships.