Practice problems for these concepts can be found at:
 Relations and Functions Solved Problems for Intermediate Algebra
 Relations and Functions Supplementary Problems for Intermediate Algebra
The Cartesian Product is basic to the mathematical concepts of relations and functions.
Definition 1. A × B = {(a, b)  a A and b B} is called the Cartesian Product.
The Cartesian Product of two sets is the set of all ordered pairs for which the first element is a member of the first set and the second element is a member of the second set.
Definition 2. A relation is any set of ordered pairs. The set of first elements is the domain and the set of second elements is the range.
A relation of real numbers is a subset of R × R (where R symbolizes the set of real numbers). A graph is a relation displayed on a coordinate system.
EXAMPLE 1. Relations (in different formats):
Definition 3. A function is a set of ordered pairs, or A relation, for which no two distinct ordered pairs have the same first element.
Definition 3. (Alternate) A function from a set A to a set B is a mapping, rule, or correspondence that assigns to each element x in set A exactly one element y in set B. (Each x is mapped to a unique y.)
A function is a relation with an added condition. The added condition is that for a function, no x value can have more than one y value associated with it.
EXAMPLE 2. In Example 1 above, (a) is a function, and (b) and (c) are not functions.
See solved problem 8.2.
To determine whether a relation in graphical format is a function, as in solved problem 8.2 (c) and (d), we can use the Vertical Line Test.
VERTICAL LINE TEST: A graph is the graph of a function if no vertical line intersects the graph in more than one point.
If any vertical line intersects a graph in two or more points, then the graph is not the graph of a function. The reason is this. Two points on the same vertical line indicate two different ordered pairs that have the same first element. Thus, it can't be a function.
A function is frequently described using set builder notation as in {(x, y)  y = 3x – 5}. In this form, the first variable in the ordered pair is called the independent variable and the second variable is the dependent variable (since its value depends on the value of the first variable).
The definition of a function indicates that it is a set of ordered pairs. To distinguish between two or more functions, we can give each a name, as in f = {(x, y)  y = x^{2} + 3x} and g = {(x, y)  y = x/(x – 2)}. In addition, to be more efficient, a more compact notation has been created to designate functions. The function f above would be designated as f (x) = x^{2} + 3x, and g would be g (x) = x/(x – 2). (Everything previously stated about functions is still valid for this notation.) "f (x)" is read "f of x" or "f evaluated at x." This notation does not imply multiplication even though it looks like it should. This is simply new notation. One advantage of this notation is its simplicity: e.g., f(x) above could be used anywhere x^{2} + 3x would be used. The independent variable x in this notation is just a placeholder for a number or expression. For this function, the symbolism "f (5)" means to evaluate the expression "x^{2} + 3x" at x = 5. That is, f (5) = (5)^{2} + 3(5). Another way of thinking of this function f is as f (?) = (?)^{2} + 3 (?) where each "?" is reserving space for a number or expression. To evaluate the function for a particular number or expression, replace each occurrence of the independent variable (or the "?") by that number or expression. Use parentheses around the number or expression to avoid confusion and to retain the correct order of operations in the result.

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