What Types of Things Do Functions Process?

Here are some examples:

Set of even numbers: {..., -4, -2, 0, 2, 4, ...}
Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, 19, 23, ...}
Positive multiples of 2 that are less than 10: {2,4, 8}

Each individual thing in the set (such as "4") is called a member, or element.

So, a function takes elements of a set, and gives back elements of a set.

But a function has special rules:

• It must work for every possible input value
• And it has only one relationship for each input value
• This can be said in one definition:
• A function relates each element of a set
  with exactly one element of another set
  (possibly the same set).
• "...each element..." means that every element in X is related to some element in Y.
  We say that the function covers X (relates every element of it).
  (But some elements of Y might not be related to at all, which is fine.)
  
  
  "...exactly one..." means that a function is single valued. It will not give back 2 or more results for the same input.

Conclusion

• a function relates inputs to outputs
• a function takes elements from a set (the domain) and relates them to elements in a set (the codomain).
• all the outputs (the actual values related to) are together called the range
• a function is a special type of relation where:
  • every element in the domain is included, and
  • any input produces only one output (not this or that)
• an input and its matching output are together called an ordered pair
• so a function can also be seen as a set of ordered pairs