Relation
In mathematics, a relation is a set of ordered pairs. Think of it as a collection of links between two sets of items. Each ordered pair contains a first element, known as the x-coordinate, and a second element, known as the y-coordinate. For example, the relation we are looking into includes the pairs \((-2, 5)\), \((-1, 3)\), \((3, 7)\), and \((4, 12)\). This simply means that there is a connection between -2 and 5, -1 and 3, and so on. It is important to understand that relations can be thought of as mappings or correspondences between elements of two sets.
Function
A function is a special type of relation. For a relation to be a function, every x-coordinate (first element of the pair) must be paired with exactly one unique y-coordinate (second element of the pair). In simple terms, there can be no x-coordinate that is linked to more than one y-coordinate. In our example, each x-coordinate \(-2, -1, 3, 4\) is paired with exactly one unique y-coordinate. Hence, the relation \{(-2, 5), (-1, 3), (3, 7), (4, 12)\} is indeed a function. This uniqueness is what differentiates functions from general relations.
x-coordinates
The x-coordinates in a relation are the first numbers in each of the ordered pairs. These values make up the domain of the relation. In our given relation \{(-2, 5), (-1, 3), (3, 7), (4, 12)\}, the x-coordinates are \(-2, -1, 3, and 4\). We can list them as a set: \(\text{Domain} = \{-2, -1, 3, 4\}\). When identifying x-coordinates, we are essentially identifying all possible input values within the given pairs. Each unique x-coordinate in a function corresponds to a single specific y-coordinate, maintaining the integrity of the function.
y-coordinates
The y-coordinates in a relation are the second numbers in each of the ordered pairs. These values make up the range of the relation. For the relation \{(-2, 5), (-1, 3), (3, 7), (4, 12)\}, the y-coordinates are \(5, 3, 7, and 12\). We can list them as a set: \(\text{Range} = \{5, 3, 7, 12\}\). Identifying y-coordinates helps us understand all possible output values that result from plugging in x-coordinates into the relation's mapping. In a function, each y-coordinate is uniquely linked to one specific x-coordinate, making it simple to predict the output for any given input.
