Algebra 1 Section 5.2

Definition In mathematics, a relation is an association between two sets. Specifically, any set of ordered pairs is a relation.

Relations Relations can be represented in a variety of ways: Listed as sets of ordered pairs Tables Graphs

Definitions The domain of a relation is the set of first elements (x-coordinates) of the ordered pairs. The range of a relation is the set of second elements (y-coordinates) of the ordered pairs.

Example 1 A = {(-3, 5), (2, 2), (2, 4), (4, -1)} Domain: D = {-3, 2, 4} Range: R = {-1, 2, 4, 5}

Example 1 A mapping diagram of relation A would look like this: -1 -3 Domain Range -1 2 4 5 -3 2 4

Definition A function is a relation in which each x-coordinate is paired with one and only one y- coordinate.

Functions Functions are relations. Each function has a domain and a range. Each function can be illustrated with ordered pairs, a table, a graph, or a mapping diagram. Not all relations are functions!

Example 3 A = {(-3, 5), (2, 2), (2, 4), (4, -1)} A is not a function. The domain element 2 if paired with two different members of the range.

Example 3 B = {(0, 4), (-1, 2), (3, 4), (-4, -3)} B is a function. Each member of the domain is paired with one and only one member of the range.

Functions Lowercase letters, particularly f and g, are used to name functions. f = {(6, 9), (4, 12), (0, 4)}

Definitions The variable that represents the values of the domain, typically x, is called the independent variable.

Definitions Because the value of the second coordinate, usually y, depends on the value chosen for the first coordinate, it is called the dependent variable.

Functional Notation The y-coordinate is symbolized as f(x), read “f of x.” Since y = f(x), the ordered pair (x, y) can be written (x, f(x)).

Functional Notation f = {(6, 9), (4, 12), (0, 4)} f(6) = 9 f(4) = 12

Example 4 Are these ordered pairs a relation? Yes; any set of ordered pairs is a relation.

Example 4 What is the domain? D = {0, 1, 2, ... , 12} These are the values of the independent variable.

Example 4 What is the range? R = {0, 2, 4, ... , 16} These are the values of the dependent variable.

Example 4 Are these ordered pairs a function? Yes, it is a function. No element of the domain relates to more than one element of the range.

Homework: pp. 190-192