In its simplest form the domain is all the values that go into a function, and the range is all the values that come out.

But in fact they are very important in **defining** a function. Read on!

*Please read What is a Function? first.*

## Functions

A function * relates* an input to an output:

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) = 200 cm

Saying "** h(10) = 200**" is like saying 10 is related to 200. Or 10 → 200

## Input and Output

But not all values may work!

- The function may not work if we give it the wrong values (such as a negative age),
- And knowing the values that can come out (such as always positive) can also help

So we need to say all the values that **can go into** and **come out of** a function.

This is best done using** Sets ...**

### A set is a collection of things, such as numbers.

Here are some examples:

Set of even numbers: {..., -4, -2, 0, 2, 4, ...}

Set of odd numbers: {..., -3, -1, 1, 3, ...}

Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}

Positive multiples of 3 that are less than 10: {3, 6, 9}

This is best done using** Sets ...**

In fact, a function is defined in terms of sets:

## Formal Definition of a Function## A function relates each element of a set |

## Domain, Codomain and Range

There are special names for **what can go into**, and **what can come out** of a function:

What can go into a function is called the Domain | |

What may possibly come out of a function is called the Codomain | |

What actually comes out of a function is called the Range |

### Example

• The set "A" is the **Domain**,

• The set "B" is the **Codomain**,

• And the set of elements that get pointed to in B (the actual values produced by the function) are the **Range**, also called the Image.

And we have:

- Domain: {1, 2, 3, 4}
- Codomain: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
- Range: {3, 5, 7, 9}

## Part of the Function

Now, what comes **out** *(the Range)* depends on what we put **in** *(the Domain)* ...

... but **WE** can define the Domain!

In fact the Domain is an essential part of the function. Change the Domain and we have a different function.

Example: a simple function like f(x) = x^{2} can have the **domain** (what goes in) of just the counting numbers {1,2,3,...}, and the **range** will then be the set {1,4,9,...}

And another function g(x) = x^{2} can have the domain of integers {...,-3,-2,-1,0,1,2,3,...}, in which case the range is the set {0,1,4,9,...}

Even though both functions take the input and square it, they have a In this case the range of g(x) also includes 0. | |

Also they will have different properties. For example f(x) always gives a unique answer, but g(x) can give the same answer with two different inputs (such as |

So, the domain is an essential part of the function.

## Does Every Function Have a Domain?

Yes, but in simpler mathematics we never notice this, because the domain is ** assumed**:

- Usually it is assumed to be something like "all numbers that will work".
- Or if we are studying whole numbers, the domain is assumed to be whole numbers.
- etc.

But in more advanced work we need to be more careful!

## Codomain vs Range

The Codomain and Range are both on the output side, but are subtly different.

The Codomain is the set of values that could **possibly** come out. The Codomain is actually **part of the definition** of the function.

And The Range is the set of values that **actually do** come out.

Example: we can define a function * f(x)=2x* with a domain and codomain of integers (because we say so).

But by thinking about it we can see that the range (actual output values) is just the **even** integers.

So the codomain is integers (we defined it that way), but the range is even integers.

The Range is a subset of the Codomain.

**Why both?** Well, sometimes we don't know the * exact* range (because the function may be complicated or not fully known), but we know the set it

*(such as integers or reals). So we define the codomain and continue on.*

**lies in**## The Importance of Codomain

Let me ask you a question: Is ** square root** a function?

If we say the codomain (the possible outputs) is **the set of real numbers**, then square root is **not a function**! ... is that a surprise?

The reason is that there could be two answers for one input, for example ** f(9) = 3** or

*-3*A function must be * single valued*. It cannot give back 2 or more results for the same input. So "f(9) = 3

**or**-3" is not right!

But it can be fixed by simply **limiting the codomain** to non-negative real numbers.

√In fact, the radical symbol (like **√x**) always means the principal (positive) square root, so **√x** is a function because its codomain is correct.

So, **what we choose for the codomain** can actually affect whether something is a **function or not.**

## Notation

Mathematicians don't like writing lots of words when a few symbols will do. So there are ways of saying "the domain is", "the codomain is", etc.

This is the neatest way I know:

this says that the function " N" (the natural numbers), and a codomain of "N" also. | |

| and either of these say that the function "f" takes in "x" and returns "x |

There is also:

Dom(f) or Dom f meaning "the domain of the function f"

Ran(f) or Ran f meaning "the range of the function f"

## How to Specify Domains and Ranges

Learn how to specify Domains and Ranges at Set Builder Notation.

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What is a Function? Injective, Surjective and Bijective Introduction to Sets Sets Index