what is a function

To tell the difference between a relationship that is a function and one that isn't:

In a function, each x-value has only one y-value paired with it.

For instance,  y = x² is a function because any x-value that you choose has only one y-value paired with it:

           when  x = 2, y = 4         when x = -2, y = 4       when x = 0, y = 0      when x = 12, y = 144    etc.

    as points, these are:  (2, 4)                (-2, 4)                         (0, 0)                      (12, 144)    

           (using the same y-value twice is OK).

But  y²  =  x  is not a function because 

           when  x = 4, y can be either 2 or -2      

     as points, these are  (4,2)    (4, -2)            (using the same x-value twice isn't OK).

a function is a relationship or expression involving one or more variables

Input, Relationship, Output

We will see many ways to think about functions, but there are always three main parts:

• The input
• The relationship
• The output

Some Examples of Functions

But we are not going to look at specific functions ...
... instead we will look at the general idea of a function.

Names

First, it is useful to give a function a name.

The most common name is "f", but we can have other names like "g" ... or even "marmalade" if we want.

But let's use "f":

We say "f of x equals x squared"

what goes into the function is put inside parentheses () after the name of the function:

So f(x) shows us the function is called "f", and "x" goes in

And we usually see what a function does with the input:

f(x) = x² shows us that function "f" takes "x" and squares it.

The "x" is Just a Place-Holder!

Don't get too concerned about "x", it is just there to show us where the input goes and what happens to it.

It could be anything!

Sometimes There is No Function Name

Sometimes a function has no name, and we see something like:

y = x²

But there is still:

• an input (x)
• a relationship (squaring)
• and an output (y)

Relating

At the top we said that a function was like a machine. But a function doesn't really have belts or cogs or any moving parts - and it doesn't actually destroy what we put into it!

A function relates an input to an output.

Saying "f(4) = 16" is like saying 4 is somehow related to 16. Or 4 → 16

What Types of Things Do Functions Process?

	... which numbers? For example, the tree-height function h(age) = age×20 makes no sense for an age less than zero.
	... it could also be letters ("A"→"B"), or ID codes ("A6309"→"Pass") or stranger things.

So we need something more powerful, and that is where sets come in:

A set is a collection of things. Here are some examples: Set of even numbers: {..., -4, -2, 0, 2, 4, ...} Set of clothes: {"hat","shirt",...}  Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...} Positive multiples of 3 that are less than 10: {3, 6, 9}

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

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

A Function is Special

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:

Formal Definition of a Function A function relates each element of a set with exactly one element of another set (possibly the same set).

The Two Important Things!

Note: "One-to-many" is not allowed, but "many-to-one" is allowed:
(one-to-many)		(many-to-one)
This is NOT OK in a function		But this is OK in a function

When a relationship does not follow those two rules then it is not a function ... it is still arelationship, just not a function.

Vertical Line Test On a graph, the idea of single valued means that no vertical line ever crosses more than one value. If it crosses more than once it is still a valid curve, but is not a function.

Some types of functions have stricter rules, to find out more you can read Injective, Surjective and Bijective

Infinitely Many

My examples have just a few values, but functions usually work on sets with infinitely many elements.

Domain, Codomain and Range

In our examples above

• the set "X" is called the Domain,
• the set "Y" is called the Codomain, and
• the set of elements that get pointed to in Y (the actual values produced by the function) is called the Range.

We have a special page on Domain, Range and Codomain if you want to know more.

So Many Names!

Functions have been used in mathematics for a very long time, and lots of different names and ways of writing functions have come about.

Here are some common terms you should get familiar with:

Ordered Pairs

And here is another way to think about functions:

Write the input and output of a function as an "ordered pair", such as (4,16).

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) )

Set of Ordered Pairs

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

But the function has to be single valued, so we also say

"if it contains (a, b) and (a, c), then b must equal c"

Which is just a way of saying that an input of "a" cannot produce two different results.

A Benefit of Ordered Pairs We can graph them... ... because they are also coordinates! So a set of coordinates is also a function (if they follow the rules above, that is)

A Function Can be in Pieces

We can create functions that behave differently depending on the input value

Example: A function with two pieces:

• when x is less than 0, it gives 5,
• when x is 0 or more it gives x²

Here are some example values:  x y -3 5 -1 5 0 0 2 4 4 16 ... ...

Explicit vs Implicit

One last topic: the terms "explicit" and "implicit".

"Explicit" is when the function shows us how to go directly from x to y, such as:

That is the classic y = f(x) style.

"Implicit" is when it is not given directly such as:

It may be hard (or impossible!) to go directly from x to y.