Functions

Functions? Functions deserve a page of their own? Well, they actually deserve more: functions are fundamental to mathematics, the other sciences and how we view the world, and also are the single most important concept you'll learn in this course. In fact, this course could be called "Introduction to Functions", with Math 1120 being called "Analysis of Functions".

So, what is a function and why are they so important? Officially a function is a triple consisting of

1. one set which we will call D,
2. a second set which we will call C
3. a rule f which associates a unique element of C to each element of D.

The important part of the definition is not the sets involved, but the rule. That rule is the function.

Let's look at the definition more carefully: it says that if you give me an element in D, the rule spits out precisely one element in C. In some sense, you can imagine the element in D as being the cause of the associated element in C. Whenever we look for causes of events, we really are assuming that the event is a function of the cause, or that whenever the cause is present, the event will occur.

Let's look for some unusual functions:

• Mixing Colours Imagine yourself as an painter, painting the northern lights on Ellesmere Island. You open your painting box only to discover that the only colours you have are red, yellow and blue. How can you get that unusual green hue? At this point you thank your lucky stars that you have read this page, and know that mixing colours is a function. D is the set of colours "red, yellow, blue", C is the set of all colours and f is the rule "add yellow" (say). After adding yellow red becomes orange, yellow stays yellow and blue becomes green. If "add yellow" was not a function you could spend your entire life on Ellesmere Island mixing yellow and blue and never getting green.
• Fingerprints As we all know, fingerprints are supposed to identify the person uniquely. D is the set of all fingerprints of all people who have ever lived, C is the set of all people who have ever lived and f associates each person to his (or her) fingerprints. If we did not have a function, two people could have the same fingerprints and you could find yourself convicted of a crime soloely because you and the actual criminal had the same fingerprints.
• Diagnosing Problems When you go to your family doctor, you describe the symptoms of your malady, he (or she) does a physical examination, takes samples, and eventually makes a pronouncement: "You have ....." The assumption behind diagnosis is that each ailment produces a particular set of symptoms. So, D is the set of ailments, C is the collection of all possible sets of symptoms and f associates the set of symptoms to each ailment. Could you imagine trying to diagnose illnesses if each ailment produced a completely random set of symptoms?

What makes each of these examples functions is that only one "result" is associated to each "cause". The last example is particularly interesting since diagnosing ailments really involves trying to invert the function: your doctor tries to identify the ailment from the symptoms, or, in other words, tries to associate the ailment to the symptoms. Since different ailments do have the same symptoms, this is not a function. Diagnosis often involves identifying a possible set of ailments, and then identifying the most likely one.