f(x) 
“f(x) = … ” is the classic way of writing a function. And there are other ways, as you will see! 
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
Input Relationship Output0  × 2  0 
1  × 2  2 
7  × 2  14 
10  × 2  20 
…
You are watching: The function of the "ciliary escalator" is to 
…  … 
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.
Example: with f(x) = x2:
an input of 4 becomes an output of 16.
In fact we can write f(4) = 16.
The “x” is Just a PlaceHolder!
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!
So this function:
f(x) = 1 – x + x2
Is the same function as:
f(q) = 1 – q + q2 h(A) = 1 – A + A2 w(θ) = 1 – θ + θ2
The variable (x, q, A, etc) is just there so we know where to put the values:
f(2) = 1 – 2 + 22 = 3
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
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 × 200  0 
1  20 
3.2  64 
15  300 
…  … 
What Types of Things Do Functions Process?
“Numbers” seems an obvious answer, but …
… which numbers? For example, the treeheight 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 SpecialBut 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 FunctionA function relates each element of a setwith exactly one element of anotherset(possibly the same set). The Two Important Things!
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: The relationship x → x2Could also be written as a table: X: x Y: x2
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: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 has no relationship does not matter) Vertical Line TestOn 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 ManyMy examples have just a few values, but functions usually work on sets with infinitely many elements.
Example: y = x3The output set “Y” is also all the Real Numbers We can”t show ALL the values, so here are just a few examples: X: x Y: x3
Domain, Codomain and RangeIn 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:
Example: z = 2u3:“u” could be called the “independent variable” “z” could be called the “dependent variable” (it depends on the value of u) Example: f(4) = 16:“4” could be called the “argument””16” could be called the “value of the function” Example: h(year) = 20 × year:h() is the function”year” could be called the “argument”, or the “variable”a fixed value like “20” can be called a parameter We often call a function “f(x)” when in fact the function is really “f” Ordered PairsAnd 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) )
Example: (4,16) means that the function takes in “4” and gives out “16” Set of Ordered PairsA 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) 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. Example: {(2,4), (2,5), (7,3)} is not a function because {2,4} and {2,5} means that 2 could be related to 4 or 5. In other words it is not a function because it is not single valued A Benefit of Ordered PairsWe 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 PiecesWe 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 3
Read more at Piecewise Functions. Explicit vs ImplicitOne last topic: the terms “explicit” and “implicit”. Explicit is when the function shows us how to go directly from x to y, such as:
y = x3 − 3 When we know x, we can find y That is the classic y = f(x) stylethat we often work with. Implicit is when it is not given directly such as:
x2 − 3xy + y3 = 0 When we know x, how do we find y? It may be hard (or impossible!) to go directly from x to y. See more: Is Marijuana Legal In Aruba ? “Implicit” comes from “implied”, in other words shown indirectly. GraphingConclusiona 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 rangea 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 pairso a function can also be seen as a set of ordered pairs 