1.3.1 Relations and Functions

The technical term for the relationships we have considered so far is relation. We use this term whenever values of one variable are paired with values of another variable, whether or not the two variables appear to be related.

Let's consider the relation that pairs the price you pay for an airplane ticket and the distance that you fly. Although one might think that the cost of a ticket should depend on how far you plan to fly, it often happens that two tickets for flights of the same length (even on the same plane!) have different prices. The price of a ticket is not uniquely determined by the length of the flight. Given a value for the independent variable (distance), there is not just one value for the dependent variable (price).

We contrast this example with the cigarette consumption relation from Activity 1 in the Section 1.2. As we have seen already, in each tabulated year there is a definite number representing the per capita consumption in the U.S. Thus, for each value of the independent variable time (a calendar year), there is just one value of the dependent variable, consumption. There are many important relations like this, in which each value of the independent variable is paired with a unique value for the dependent variable. Whenever this is true, the relation is called a function.

In the airplane ticket relation, we are not guaranteed a unique value for the dependent variable, so this relation is not a function.

Your fitted line for the cigarette consumption relation (Activity 1 in the preceding section) provides a second example of a function. That line has a formula of the form y = mx + b, where m and b are whatever numbers you calculated to make the line fit the data. For each value of the independent variable x (a year), there is a unique value for the dependent variable y, the predicted per capita consumption. For integer values of x between 1975 and 2000, these y-values should be very similar to the actual per capita consumption, but this formula-based function is not the same function as the number pairing represented by actual consumption.

We hasten to add that it is not having a formula such as y = mx + b that makes a relation a function; rather, it is the pairing of values of the independent variable with unique values of the dependent variable. We have to be aware of the distinction between functions that are defined by data (or graphs of data) and functions that are defined by formulas. The latter are often useful models of reality, but they are not in fact that reality.

We will often find it useful to think of a function as a process that associates each permissible first coordinate with a unique second coordinate — a pairing process, with the special condition that the second value in each pair is uniquely determined by the first. In this course, it is important to think about functions as being dynamic, as “doing something” to an x-value to get a y-value. What a function does to each value of the independent variable is called a rule for the function. For example:

• If a function is defined by a two-column table of data, the dynamic interpretation (or rule) of the function is, “For each number in the left column, look for the paired number in the right column.”
• If a function is defined by a graph, the rule is, “For each first coordinate on the horizontal axis, go up to the corresponding point on the graph, and then locate the second coordinate on the vertical axis.”
• For the model function defined by the formula `y=-2.77 x+5555`, the dynamic interpretation is, “For each number `x`, multiply `x` by `-2.77`, and then add `5555` to get the corresponding `y`.”

The “For each number” parts of the preceding examples bring us to another important characteristic of a function: the set of numbers that are allowed to be values of the independent variable. This set is called the domain of the function. Thus, if a function is defined by a table, its domain is the set of numbers in the left-hand column. If it is defined by a graph, its domain is the set of first coordinates of points on the graph. If it is defined by a formula, the domain is the set of numbers that can legitimately be substituted for the independent variable in the formula.

By combining the ideas of domain and rule, we have another way — in addition to the pairing definition — to characterize the concept of function:

1. We start with a collection of numbers that constitute allowable inputs to the function. This set is the domain of the function.
2. We have a rule for associating with each input a unique output or result or value of the function.

The connection between the domain/rule description and the pairing definition is this: The domain identifies which numbers can be first elements of a pair, and the rule tells us what number is the second element that goes with each first element.

Checkpoint 1

Contents for Chapter 1