Functions are very specific types of relations. Before defining a function, it is important to define a relation.

#### Relations

Any set of ordered pairs is called a **relation.** Figure 1 shows a set of ordered pairs.

*A* = {(–1, 1), (1, 3), (2, 2), (3, 4)}

#### Domain and range

The set of all *x*'s is called the domain of the relation. The set of all *y*'s is called the range of the relation. The domain of set *A* in Figure 1 is {–1, 1, 2, 3}, while the range of set *A* is {1, 2, 3, 4}.

##### Example 1

Find the domain and range of the set of graphed points in Figure 2.

The domain is the set {–2, –1, 1, 3}. The range is the set {–1, 2, 3}.

#### Defining a function

The relation in Example
*x* value of each pair of coordinates is different, the relation is called a *function.* A function is a relation in which each member of the domain is paired with exactly one element of the range. *All functions are relations, but not all relations are functions.* A good example of a functional relation can be seen in the linear equation *y* = *x* + 1, graphed in Figure 3. The domain and range of this function are both the set of real numbers, and the relation is a function because for any value of *x* there is a unique value of *y.*

*y*=

*x*+ 1.

#### Graphs of functions

In each case in Figure 4 (a), (b), and (c), for any value of *x*, there is only one value for *y*. Contrast this with the graphs in Figure 5.

#### Graphs of relationships that are not functions

In each of the relations in Figure 5 (a), (b), and (c), a single value of *x* is associated with two or more values of *y*. These relations are not functions.

#### Determining domain, range, and if the relation is a function

##### Example 2

Note that Examples 2(e) and (f) are illustrations of *inverse relations*: relations in which the domain and the range have been interchanged. Notice that while the relation in (e) is a function, the inverse relation in (f) is not.

#### Finding the values of functions

The *value of a function* is really the *value of the range* of the relation. Given the function

*f =* {(1, –3), (2, 4), (–1, 5), (3, –2)}

the value of the function at 1 is –3, at 2 is 4, and so forth. This is written *f*(1) = –3 and *f*(2) = 4 and is usually read, “ *f* of 1 = –3 and *f* of 2 = 4.” The lowercase letter *f* has been used here to indicate the concept of function, but any lowercase letter might have been used.

##### Example 3

Let *h* = {(3, 1), (2, 2), (1, –2), (–2, 3)} Find each of the following.

##### Example 4

If *g*( *x*) = 2 *x* + 1, find each of the following.

##### Example 5

If *f* ( *x*) = 3 *x* ^{2} + *x* –1, find the range of *f* for the domain {–2, –1, 1}.