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!

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

Example: "Multiply by 2" is a very simple function.

Here are the three parts:

Input Relationship Output
0 × 2 0
1 × 2 2
7 × 2 14
10 × 2 20
... ... ...

For an input of 50, what is 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 we can have other names like "g" ... or even "marmalade" if we want.

But let's use "f":

f(x)

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 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!

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:

age h(age) = age × 20
0 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 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!

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!

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

function   function
(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 a relationship, just not a function.

Example: The relationship x → x2

function

Could also be written as a table:

X: x Y: x2
3 9
1 1
0 0
4 16
-4 16
... ...

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 has no relationship does not matter)

 

not single valued

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.

Example: y = x3

  • The input set "X" is all Real Numbers
  • The 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
-2 -8
-0.1 -0.001
0 0
1.1 1.331
3 27
and so on... and so on...

 

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:

Function Parts

Example: with z = 2u3:

  • "u" could be called the "independent variable"
  • "z" could be called the "dependent variable" (it depends on the value of u)

Example: with f(4) = 16:

  • "4" could be called the "argument"
  • "16" could be called the "value of the function"

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) )

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)

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

 

interactive-cartesian-coordinates  

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

Read more at Piecewise Functions.

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:

y = x3 - 3

When we know x, we can find y

That is the classic y = f(x) style.

"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.

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

Graphing

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