Introduction to Functions
by mrichard, Oct 30, 2018, 4:51 AM
Previously, we talked about sets. Once we have these basic objects, we can start to create, define, and communicate relationships and patterns between sets. Mathematics uses the language of a function to describe these relationships.
Most simply, a function is a rule that takes elements on one set (often called inputs) and assigns them to elements of another set (often called outputs.) The one rule to remember to make sure you have a proper function is a function only has a single output for each input.
There are many notations used to discuss functions. The most common, and the one used throughout this blog, will be the form
. We call 
the function, where it takes elements from the set 
and assigns them to elements in the set 
.
So, we could call the function from our example
While the letter used to name the function can be arbitrary, it is common to use , 
, or 
if no other information is given.
It is good to have a common language to describe the sets and in the context of a function.
Let's consider the function
from before. We say that 
is the domain, and 
is the codomain.
Imagine you wanted to tell a friend that our function took the input 
and assigned it to strawberries; however, you don't want to copy down the list of rules we had each time! This may not seem like a big deal, but imagine you had a domain with 
elements. We'd need an easier notation for these situations!
The accepted notation to tell our friends that sends to 
is
This is most commonly read as " of equals 
"
This notation is very flexible, as it allows us to have more than one element as the input. For example, since the subset
has elements assigned to 
we can say
Remember that a set does not have its elements in a specific order, so we can also write
Given just the above, we cannot tell exactly what 
and 
are; if we need to communicate that information, we can list them. But there are many situations where you just need to know if a particular element (very frequently 
or 
) is among the outputs of a set of inputs.
As noted before, given the function
, there is no guarantee that every element in is actually an output of the function! We are only guaranteed that contains every output. So, it is then natural to wonder what exactly 
is. Since is the domain (i.e. the set of all possible inputs) of , every element in is assigned to some element in . So, must be a subset of , and in particular is the set that contains all possible outputs of the functions, and no extra elements.
Since all possible outputs must exist in the codomain, for any function we must have 
It is also good to note that a function can assign different inputs to the same output!
Questions
Consider the function
defined by 
and 
Consider another function
.
Consider a function
Suppose 
Most simply, a function is a rule that takes elements on one set (often called inputs) and assigns them to elements of another set (often called outputs.) The one rule to remember to make sure you have a proper function is a function only has a single output for each input.
and the set 
We create a function that takes elements of 
to anything. However, There are many notations used to discuss functions. The most common, and the one used throughout this blog, will be the form
. We call 
the function, where it takes elements from the set 
and assigns them to elements in the set 
.So, we could call the function from our example
While the letter used to name the function can be arbitrary, it is common to use , 
, or 
if no other information is given.It is good to have a common language to describe the sets
and in the context of a function.Let's consider the function
from before. We say that 
is the domain, and 
is the codomain.Imagine you wanted to tell a friend that our function
took the input 
and assigned it to strawberries; however, you don't want to copy down the list of rules we had each time! This may not seem like a big deal, but imagine you had a domain with 
elements. We'd need an easier notation for these situations!The accepted notation to tell our friends that
sends to 
is
This is most commonly read as " of equals 
"This notation is very flexible, as it allows us to have more than one element as the input. For example, since the subset
has elements assigned to 
we can say
Remember that a set does not have its elements in a specific order, so we can also write
Given just the above, we cannot tell exactly what 
and 
are; if we need to communicate that information, we can list them. But there are many situations where you just need to know if a particular element (very frequently 
or 
) is among the outputs of a set of inputs.
is assigned to the element 
,
and say 
is equal to 
.
is some subset of 
are assigned exactly to elements in some 
,
However, it is accepted that As noted before, given the function
, there is no guarantee that every element in is actually an output of the function! We are only guaranteed that contains every output. So, it is then natural to wonder what exactly 
is. Since is the domain (i.e. the set of all possible inputs) of , every element in is assigned to some element in . So, must be a subset of , and in particular is the set that contains all possible outputs of the functions, and no extra elements.Since all possible outputs must exist in the codomain
, for any function we must have 
,
defined by:
Then, 
So, the range of the function is 
,It is also good to note that a function can assign different inputs to the same output!
by
All three elements of 
is a valid function.Questions
Consider the function
defined by 
and 
.
.
Consider another function
.
We can choose from the three elements in 
,
to be different from 
ways so that every input is sent to a different output.Consider a function
Suppose 
Since 
elements, there are 
different ways.
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