**Introduction to Relations and Functions**

Consider the following statements. Each of them represents a situation that could occur in everyday life.

- The outdoor air temperature varies with the time of day.
- The time the sun is above the horizon on June 21 varies with the latitude of the observer.
- The time required for a wet rag to dry depends on the air temperature.

All of these expressions involve something that depends on something else. In the first case, a statement is made concerning temperature versus time; in the second case, a statement is made concerning sun-up time versus latitude; in the third case, a statement is made concerning time versus temperature. Here, the term *versus* means ''compared with.''

**Independent vs. Dependent Variables**

**Independent Variables**

An *independent variable* changes, ut its value is not influenced by anything else in a given scenario. Time is often treated as an independent variable. A lot of things depend on time.

When two or more variables are interrelated, at least one of the variables is independent, but they are not all independent. A common and simple situation is one in which there are two variables, one of which is independent. In the three situations described above, the independent variables are time, latitude, and air temperature.

**Dependent Variables**

A *dependent variable* changes, but its value is affected by at least one other factor in a situation. In the scenarios described above, the air temperature, the sun-up time, and time are dependent variables.

When two or more variables are interrelated, at least one of them is dependent, but they cannot all be dependent. Something that's an independent variable in one instance can be a dependent variable in another case. For example, the air temperature is a dependent variable in the first situation described above, but it is an independent variable in the third situation.

**Scenarios Illustrated**

The three scenarios described above lend themselves to illustration. In order, they are shown crudely in Fig. 1-1.

Figure 1-1A shows an example of outdoor air temperature versus time of day. Drawing B shows the sun-up time (the number of hours per day in which the sun is above the horizon) versus latitude on June 21, where points south of the equator have negative latitude and points north of the equator have positive latitude. Drawing C shows the time it takes for a rag to dry, plotted against the air temperature.

The scenarios represented by Figs. 1-1A and C are fiction, having been contrived for this discussion. But Fig. 1-1B represents a physical reality; it is true astronomical data for June 21 of every year on earth.

**What is a Relation?**

All three of the graphs in Fig. 1-1 represent *relations*.

In mathematics, a relation is an expression of the way two or more variables compare or interact. (It could just as well be called a relationship, a comparison, or an interaction.) Figure 1-1B, for example, is a graph of the relation between the latitude and the sun-up time on June 21.

When dealing with relations, the statements are equally valid if the variables are stated the other way around. Thus, Fig. 1-1B shows a relation between the sun-up time on June 21 and the latitude. In a relation, ''this versus that'' means the same thing as ''that versus this.'' Relations can always be expressed in graphical form.

**When is a Relation a Function?**

A *function* is a special type of mathematical relation. A relation describes how variables compare with each other. In a sense, it is ''passive.'' A function transforms, processes, or morphs the quantity represented by the independent variable into the quantity represented by the dependent variable. A function is ''active.''

All three of the graphs in Fig. 1-1 represent functions.

The changes in the value of the independent variable can, in some sense, be thought of as causative factors in the variations of the value of the dependent variable. We might re-state the scenarios this way to emphasize that they are functions:

- The outdoor air temperature is a function of the time of day.
- The sun-up time on June 21 is a function of the latitude of the observer.
- The time required for a wet rag to dry is a function of the air temperature.

A relation can be a function only when every element in the set of its independent variables has at most one correspondent in the set of dependent variables. If a given value of the dependent variable in a relation has more than one independent-variable value corresponding to it, then that relation might nevertheless be a function. But if any given value of the independent variable corresponds to more than one dependent-variable value, that relation is not a function.

**Reversing the Variables**

In graphs of functions, independent variables are usually represented by horizontal axes, and dependent variables are usually represented by vertical axes. Imagine a movable, vertical line in a graph, and suppose that you can move it back and forth. A curve represents a function if and only if it never intersects the movable vertical line at more than one point.

Imagine that the independent and dependent variables of the functions shown in Fig. 1-1 are reversed. This results in some weird assertions:

- The time of day is a function of the outdoor air temperature.
- The latitude of an observer is a function of the sun-up time on June 21.
- The air temperature is a function of the time it takes for a wet rag to dry.

The first two of these statements are clearly ridiculous. Time does not depend on temperature. You can't make time go backwards by cooling things off or make it rush into the future by heating things up. Your geographic location is not dependent on how long the sun is up. If that were true, you would be at a different latitude a week from now than you are today, even if you don't go anywhere (unless you live on the equator!).

If you turn the graphs of Figs. 1-1A and **B** sideways to reflect the transposition of the variables and then perform the vertical-line test, you'll see that they no longer depict functions. So the first two of the above assertions are not only absurd, they are false.

Figure 1-1C represents a function, at least in theory, when ''stood on its ear.'' The statement is still strange, but it can at least be true under certain conditions. The drying time of a standard-size wet rag made of a standard material could be used to infer air temperature experimentally (although humidity and wind speed would be factors too). When you want to determine whether or not a certain graph represents a mathematical function, use the vertical-line test, not the common-sense test!

**Domain and Range**

Let *f* be a function from set *A* to set *B*. Let *A'* be the set of all elements *a* in *A* for which there is a corresponding element *b* in *B*. Then *A'* is called the *domain* of *f*.

Let *f* be a function from set *A* to set *B*. Let *B'* be the set of all elements *b* in *B* for which there is a corresponding element *a*in *A*. Then *B'* is called the *range* of *f*.

**Relations and Functions Practice Problems**

**Practice 1**

Figure 1-2 is called a *Venn diagram*. It shows two sets *A* and *B*, and three points or elements *P*, *Q*, and *R*. What is represented by the cross-hatched region? Which of the points, if any, is in the intersection of sets **A** and *B*? Which points, if any, are in the union of sets *A* and *B*?

**Solution 1**

**Problem 2**

Figure 1-3 is an illustration of a relation that maps certain points in a set *C* to certain points in a set *D*. Only those points shown are involved in this relation. Is this relation a function? If so, how can you tell? If not, why not?

**Solution 2**

The relation is a function, because each value of the independent variable, shown by the points in set *C*, maps into at most one value of the dependent variable, represented by the points in set *D*.

Practice problems for these concepts can be found at:

Background Math Practice Test