Overview:
While a set of ordered pairs describes a relation, not every relation is also a function. Each x element (the domain) must be paired with only one y element (the range) to be a function. The relationship between each ordered pair can be tested to determine if the relation is actually a function.
What Are Ordered Pairs?
Imagine a graph with an x axis and a y axis. Every point on that plane can be described by its coordinates in an ordered pair. Suppose a point has coordinates (3, 1). Its position is 3 units from 0 along the x axis and 1 unit from 0 along the y axis. Another point has coordinates (2, 2). Its coordinates will be 2 units from the x axis and 2 units from the y axis.
What Is a Relation?
A relation is a set of ordered pairs. For example, one set of ordered pairs might be {(3, 1), (2, 2), (1, 3), (0, 4), (-1, 5), (-2, 6)}. Another relation may be a set of ordered pairs {(0, 5), (0, 2), (3, 1), (2, 2)}. The relation always describes points on the x axis and the y axis. A relation can always be described by its domain and range. The domain of a set of ordered pairs describe the x coordinates. In the first example, the domain would be {3, 2, 1, 0, -1, -2}. The range describes the y coordinates, or (1, 2, 3, 4, 5, 6}. In the second example, the domain would be {0, 3, 2} and the range would be {5, 2, 1, 2}.
What Is a Function?
In a function, each element in the domain is paired with one element in the range. In the example {(3, 1), (2, 2), (1, 3), (0, 4), (-1, 5), (-2, 6)}the relation is also a function. In the set of ordered pairs {(0, 5), (0, 4), (3, 1), (2, 2)} the relation is not a function because the 0 element in the domain has 2 range values of 5 and 4. Suppose a set of ordered pairs is {(1, 4), (2, 4), (3, 4), (4, 4)}. Although the domain set has the values (1, 2, 3, 4) and the range set has the value (4), it would still be a function. It isn’t a very interesting function because there is only one value in the range, but it is the only value.
What Is the Vertical Line Test?
Suppose a graph can be plotted of the relationship between 2 variables x and y. If a vertical line can be drawn through that relationship that only intersects it once, the relation is a function. If the vertical line intersects the graph of the relation more than once, then it is not a function.
Interested in algebra tutoring services? Learn more about how we are assisting thousands of students each academic year.
SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Caledonia, ON, Canada: visit Tutoring in Caledonia, ON, Canada
