Problem 8 - Algebra 2

Question

Make a mapping diagram for each relation. $$ \\{(0,0),(-1,-1),(-2,-8),(-3,-27)\\} $$

Step-by-Step Solution

Answer

The diagram includes two columns with the domain values {0, -1, -2, -3} and range values {0, -1, -8, -27}. Arrows are drawn from each domain value to the corresponding range value.

Step 1: Understand the Relation

A mapping diagram visually represents how each element from one set (domain) is paired with an element in another set (range). The given relation is a set of ordered pairs, where the first number represents an element from the domain and the second number represents the corresponding element in the range.

Step 2: Set up the Diagram

Draw two parallel columns. Label the left column 'Domain' for the set of all the first elements of the ordered pairs, and label the right column 'Range' for the set of all the second elements.

Step 3: List the Domain and Range

In the 'Domain' column, list all the unique first elements from the given ordered pairs: 0, -1, -2, -3. In the 'Range' column, list the corresponding second elements: 0, -1, -8, -27.

Step 4: Draw Arrows for the Pairs

Draw arrows from each element in the domain to its corresponding element in the range. An arrow should be drawn from 0 in the domain to 0 in the range, from -1 to -1, from -2 to -8, and from -3 to -27.

Step 5: Check for Function Criteria

In a function, each element of the domain should be paired with exactly one element of the range. Here, each element of the domain maps to a unique element in the range, consistent with the definition of a function.

Key Concepts

Relation in Algebra

In algebra, a relation is a way to show a connection or correlation between sets of values. A relation is often expressed as a set of ordered pairs, like \(\{(0,0),(-1,-1),(-2,-8),(-3,-27)\}\). Here, each first element represents an input into the relation, and the second element represents the output associated with that input.

A relation can manifest in different forms, such as a list of ordered pairs, a table, a graph, or, as illustrated by the textbook exercise, a mapping diagram. The mapping diagram is helpful for visual learners because it creates a clear picture of how each individual input has a relationship with a corresponding output.

Domain and Range

Two fundamental concepts that are essential to the understanding of relations in algebra are the domain and range. The domain of a relation is the set of all possible inputs, or in the case of ordered pairs, the set of all first elements. For the given relation, the domain is \(\{0, -1, -2, -3\}\).

On the other side, the range is the set of all possible outputs, which corresponds to the set of second elements in the ordered pairs. The range for the relation stated in the exercise is \(\{0, -1, -8, -27\}\). Whenever you encounter a set of ordered pairs, identifying the domain and range is a critical step because it sets the foundation for understanding the relationship between variables and for determining whether the relation is a function.

Function Criteria

To determine if a relation is a function, we apply specific function criteria. The rule is straightforward: each element in the domain must correspond to exactly one element in the range. This means there can’t be any repeating elements in the domain with different range values.

In the mapping diagram from the exercise, we see that each value in the domain \(\{0, -1, -2, -3\}\) points to exactly one value in the range, complying with the function criteria. If a single domain value were mapped to multiple range values, then the relation would not be a function. Understanding this criterion is vital, as functions have unique properties and uses in algebra, making the differentiation between a generic relation and a function extremely important.