The two important rules!

Let f  be a function fron the set X to the set Y,  (f:X->Y)  with value (for every element x in X) , f(x)=2x

X={1,2,3,4,5}

Y={2,4,6,8,10}

"...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.

So "f(2) = 2.2=4 or 5" is not right!

Note: "One-to-many" is not allowed, but "many-to-one" is allowed: When a

relationship does not follow those two rules then it is not a function ... it is still

a relationship, just not a function.

Example: with f(2) = 4: "2" could be

called the "argument" and "4" could be called the "value of the function".

And here is another way to think about functions:

Write the input and output of a function as an "ordered pair", such as (2,4).

They are called ordered pairs because the input always comes first, and the

output second:(input, output)So it looks like this:( x, f(x) )

A function can then be defined as a set of ordered pairs:

Example: {(1,2), (2,4), (3,6),(4,8),

(5,10)} is a function that says

"1 is related to 2", "2 is related to 4" e.t.c

A function relates inputs to outputs.

Conclusions

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.