91Ó°ÊÓ

A relation in mathematics is a collection of ordered pairs. Each ordered pair consists of two elements: a first element known as the domain element and a second known as the range element. In the given exercise, we have the set of ordered pairs \((11,6),(-1,-2),(0,0),(3,-2)\). This set is a relation because it groups elements from two sets—the domain and range—together in pairs.

Relations are useful because they provide a way to express how two sets of entities are linked.

• Each ordered pair consists of one element from the domain and one from the range.
• For example, in the pair \((11,6)\), 11 is related to 6.

The exercise asks whether this relation is also a function, which has specific rules that dictate how domain elements relate to range elements.

The domain of a relation is the set of all the first elements from each ordered pair. In simple terms, it's all the 'input' values of the relation. For the given relation \((11,6),(-1,-2),(0,0),(3,-2)\), the domain is determined by collecting all the first elements of each pair. Therefore, the domain is the set \(\{11, -1, 0, 3\}\).

Understanding the domain is crucial when working with functions because it defines all the potential inputs you can have. When evaluating whether a relation is a function, check if every element in the domain maps to exactly one element in the range. If any domain element maps to more than one range element, the relation cannot be a function.

The range of a relation, on the other hand, is the set of all second elements from each ordered pair. These are the 'output' values that result from the input values of the domain. In the relation \((11,6),(-1,-2),(0,0),(3,-2)\), the range is \(\{6, -2, 0, -2\}\).

In the context of determining whether a relation is a function, the range is important because it reflects the uniqueness of output values. A function requires that each input from the domain should map to one and only one output in the range. If the exercise asks us to determine if a relation is a function, we focus on whether each input consistently leads to a single output in the range. Despite having two elements in the range \(-2\), the relation satisfies a function's criteria because each domain element corresponds to one unique range element.