Relations and Functions Help
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
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.
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.
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.
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