Friday, January 10, 2014

What is a Function


What is a Function?

A function relates an input to an output.
It is like a machine that has an input and an output.
And the output is related somehow to the input.

  f(x)  
"f(x) = ... " is the classic way of writing a function.
And there are other ways, as you will see!


First, it is useful to give a function a name.
The most common name is "f", but you can have other names like "g" ... or even "marmalade" if you want.
But let's use "f":
f(x)
You would 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 you the function is called "f", and "x" goes in
And you will often see what a function does with the input:
f(x) = x2 shows you that function "f" takes "x" and squares it.

Example: with f(x) = x2:
  • an input of 4
  • becomes an output of 16.
In fact we can write f(4) = 16.
 Example: this tree grows 20 cm every year, so the height of the tree is related to its age using the function h:
h(age) = age × 20
So, if the age is 10 years, the height is:
h(10) = 10 × 20 = 200 cm

Here are some example values:
ageh(age) = age × 20
00
120
3.264
15300
......


A Function is Special

But a function has special rules:
  • It must work for every possible input value
  • And you can only have 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!

1.
"...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.)
2.
"...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) = 7 or 9" is not right!
function function
(one-to-many) (many-to-one)
This is NOT OK in a function But this is OK in a function
If a relationship does not follow those two rules then it is not a function ... it would still be a relationship, just not a function.

Example: The relationship x → x2

function
Could also be written as a table:
X: xY: x2
39
11
00
416
-416
......

It is a function, because:
  • Every element in X is related to Y
  • No element in X has two or more relationships
So it follows the rules.
(Notice how both 4 and -4 relate to 16, which is allowed.)

Example: This relationship is not a function:

function
It is a relationship, but it is not a function, for these reasons:
  • Value "3" in X has no relation in Y
  • Value "4" in X has no relation in Y
  • Value "5" is related to more than one value in Y
(But the fact that "6" in Y is not related to does not matter)
not single valued

Vertical Line Test

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


Ordered Pairs

I said I would show you many ways to think about functions, and here is another way:
You can 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:
xf(x) )
Example:
(4,16) means that the function takes in "4" and gives out "16"

Set of Ordered Pairs

A function can then be defined as a set of ordered pairs:
Example: {(2,4), (3,5), (7,3)} is a function that says
"2 is related to 4", "3 is related to 5" and "7 is related 3".
Also, notice that:
  • the domain is {2,3,7} (the input values)
  • and the range is {4,5,3} (the output values)

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

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