What Types of Things Do Functions Process?

A set is a collection of things

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