What do the following pairs have in common? They all describe a particular correspondence between two groups. A relation is a correspondence between two sets. The first set is called the domain, and the corresponding second set is called the range. Members of these sets are called elements.

A relation is a set of ordered pairs. The domain is the set of all the first components of the ordered pairs, and the range is the set of all the second components of the ordered pairs. A relation in which each element in the domain corresponds to exactly one element in the range is a function.

Note that the definition of a function is more restrictive than the definition of a relation. For a relation, each input corresponds to at least one output, whereas, for a function, each input corresponds to exactly one output. The blood-type example given is both a relation and a function.

Also note that the range (set of values to which the elements of the domain correspond) is a subset of the set of all blood types. However, although all functions are relations, not all relations are functions.

Study Tip  All functions are relations but not all relations are functions.

For example, at a university, four primary sports typically overlap in the late fall: football, volleyball, soccer, and basketball. On a given Saturday, the following table indicates the start times for the competitions. Because an element in the domain, 7:00 p.m., corresponds to more than one element in the range, Soccer and Basketball, this is not a function. It is, however, a relation.

Let's start with the equation , where can be any real number. This equation assigns to each exactly one corresponding . Since the variable depends on what value of is selected, we denote as the dependent variable. The variable can be any number in the domain; therefore, we denote as the independent variable.

Although functions are defined by equations, it is important to recognize that not all equations are functions.

CAUTION  Not all equations are functions.

The requirement for an equation to define a function is that each element in the domain corresponds to exactly one element in the range. Throughout the ensuing discussion, we assume to be the independent variable and to be the dependent variable. In the “equations that represent functions of ,” every corresponds to exactly one . Some ordered pairs that correspond to these functions are The fact that and both correspond to in the first two examples does not violate the definition of a function.

In the “equations that do not represent functions of ,” some correspond to more than one .

Study Tip  We say that is not a function of . However, if we reverse the independent and dependent variables, then is a function of .

Some ordered pairs that correspond to these equations are Let's look at the graphs of the three functions of : Let's take any value for , say . The graph of corresponds to a vertical line. A function of maps each to exactly one ; therefore, there should be at most one point of intersection with any vertical line. We see in the three graphs of the functions above that if a vertical line is drawn at any value of on any of the three graphs, the vertical line only intersects the graph in one place. Look at the graphs of the three equations that do not represent functions of . A vertical line can be drawn on any of the three graphs such that the vertical line will intersect each of these graphs at two points. Thus, there are two that correspond to some in the domain, which is why these equations do not define as a function of .

Study Tip  If any corresponds to more than one , then is not a function of .

We know that the equation defines as a function of because its graph is a nonvertical line and thus passes the vertical line test. We can select (input) and determine unique corresponding (output). The output is found by taking 2 times the input and then adding 5. If we give the function a name, say, “”, then we can use function notation: The symbol is read “ evaluated at ” or “ of ” and represents the that corresponds to a particular . In other words, . It is important to note that is the function name, whereas is the value of the function. In other words, the function maps some value in the domain to some value in the range. The independent variable is also referred to as the argument of a function. To evaluate functions, it is often useful to think of the independent variable or argument as a placeholder. For example, can be thought of as In other words, “ of the argument is equal to the argument squared minus 3 times the argument.” Any expression can be substituted for the argument: It is important to note:

Examples 6, 7, and 8 illustrate the following:

Now that we have shown that , we turn our attention to one of the fundamental expressions in calculus: the difference quotient. Example 9 illustrates the difference quotient, which will be discussed in detail in Section 3.2. For now, we will concentrate on the algebra involved when finding the difference quotient. In Section 3.2, the application of the difference quotient will be the emphasis.

Sometimes the domain of a function is stated explicitly. For example, Here, the explicit domain is the set of all negative real numbers, . Every negative real number in the domain is mapped to a positive real number in the range through the absolute value function.

If the expression that defines the function is given but the domain is not stated explicitly, then the domain is implied. The implicit domain is the largest set of real numbers for which the function is defined and the output value is a real number. For example, does not have the domain explicitly stated. There is, however, an implicit domain. Note that if the argument is negative, that is, if , then the result is an imaginary number. In order for the output of the function, , to be a real number, we must restrict the domain to nonnegative numbers, that is, if . In general, we ask the question, “what can be?” The implicit domain of a function excludes values that cause a function to be undefined or have outputs that are not real numbers.