Example 1: Verbal Rules

"I dress for the weather."

Here the two variable quantities are "the weather" and "what I wear." The statement "I dress for the weather" says that there is a relationship between these two quantities, although that relationship is not made explicit. If there were a rule in mind which was used to uniquely determine what you would wear for particular weather conditions, then that rule would be a function. We could write  d = f(w) , where  d  represents the variable quantity "what I wear" (d  for "dress"), w  represents the variable quantity "the weather," and  f  represents the explicit rule which determines the value of  d  from the value of  w .

Let's try to be a little more explicit ...

"If it's raining I wear my black shoes; otherwise I wear either my red shoes or my blue shoes."

This is much like the previous rule, except here the broad variables "the weather" and "what I wear" have been restricted to the much more explicit "rain or not rain" and "black shoes or red shoes or blue shoes," respectively. The rule  f  relating these variables is also more explicit:

Here  w  varies between the two values "rain" and "not rain," and  d = f(w)  varies between the three values "black," "red," and "blue."

Notice, however, that  f  does not define a function. Why?