Determining Relations and Functions
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Relation:  A relation is simply a set of ordered pairs.


A relation can be any set of ordered pairs.
No special rules need apply.
The following is an example of a relation:
relation {(1,2),(2, 4),(3, 5),(2, 6),(1, -3)}

The graph at the right shows that a vertical line may intersect more than one point in a relation.


The graph of this orange arrow is also a relation.


If we impose the following rule on a relation, it becomes a function.

Function:  A function is a set of ordered pairs in which each x-element has only ONE y-element associated with it.


The relation shown above is NOT a function because the x-element 2 is paired with a y-element of 4 and also a y-element of 6.  Similarly, the x-element of 1 is paired with the y-elements of 2 and -3.

The relation above can be altered to become a function by removing the ordered pairs where the x-coordinate is used twice. 
function:  {(1,2), (2,4), (3,5)}

The graph at the right shows that a vertical line intersects only ONE point in a function.
This is called the vertical line test for functions.

 



Just remember:
A function may not have two y-values assigned to the same x-value, such as {(2,4), (2,6)}.

A function may, however, have two x-values assigned to the same y-value, such as {(2,4), (3,4)}.
 

 

Let's take one last look at relations and functions.
Consider the following relations.  Are they also functions?

Situation 1

   Chico        Lynn        Paul   
Consider the relation described as:
(x, y) = (student's name, shirt color)
This relation consists of
{(Chico, gray), (Lynn, gray), (Paul, gray)}
This relation is also a function!
 
Now, let's reverse the situation:
(x, y) = (shirt color, student's name)
This relation consists of
{(gray, Chico), (gray, Lynn), (gray, Paul)}
This relation is NOT a function!

If you are told that the student wearing the gray shirt wants to ask you a question, how will you know which student to approach?
 

 

Situation 2

      Chico        Lynn        Paul   
Consider the relation described as:
(x, y) = (student's name, shirt color)
This relation consists of
{(Chico, gray), (Lynn, green), (Paul, red)}
This relation is also a function!
 
Now, let's reverse the situation:
(x, y) = (shirt color, student's name)
This relation consists of
{(gray, Chico), (green, Lynn), (red, Paul)}
This relation is also a function!

In this situation, if you are told that the student wearing the gray shirt wants to ask you a question, you will know that the student is Chico.
 

 

Functional Notation:

Traditionally, functions are referred to by the notation f (x), which is read "f of x" or
"f  as a function of x".  (The parentheses do not mean "multiplication".)
For example, since y = 3x + 7 is a function, it may also be written as f (x) = 3x + 7.

 The letter f need not be the only letter used, however, in function names.  Function names may also be g(x), h(x), A(x), C(a), or any letters that clearly identify the function being used. 


Example:   A function is represented by f (x) = 2x + 5.    Find f (3).

                To find f (3), replace the x-value with 3.     f (3) = 2(3) + 5 = 11.
                The answer, 11, is called the image of 3 under  f (x).

Note:  The f (x) notation can be thought of as another way of representing the y-value in a function, especially when graphing.  The y-axis if even labeled as the f (x) axis, when graphing.