Relations and Functions Help (page 2)

By — McGraw-Hill Professional
Updated on Aug 26, 2011

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.

Relations and 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!

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