What is a Function?

A function is like a machine: it has an input and an output. And the output is related somehow to the input.

Input, Relationship, Output

I will show you many ways to think about functions, but there will always be three main parts:

• The input
• The relationship
• The output

Some Examples of Functions

x² (squaring) is a function
x³+1 is also a function
Sine, Cosine and Tangent are functions used in trigonometry
and there are lots more!

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 you can have other names like "g" ... or even "marmalade" if you want.

But let's use "f":

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.

The "x" is Just a Place-Holder!

Don't get too concerned about "x", it is just there to show you 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 you might just see something like:

y = x²

But there is still:

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

Relating

At the top I 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 you 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: age h(age) = age × 20 0 0 1 20 3.2 64 15 300 ... ...

What Type 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 (usually different) elements of a set.

A Function is Special

But a function is, in fact, a special type of relationship that has special rules:

• It must work for every value in X
• And you can only have one relationship for each element in X.

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

Let us look at the two rules in detail:

(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.

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.

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

You can also 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) )

And 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

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

Read more at Piecewise Functions.

Explicit vs Implicit

Before I finish, I would like to mention the terms "explicit" and "implicit".

"Explicit" is when the function shows you 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.

"Implicit" comes from "implied", in other words shown indirectly.

Graphing

• My Function Grapher can only handle explicit functions,
• But my Equation Grapher can handle both types (but takes a little longer).

Conclusion

That is All for Now

Happy functioning!