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 xelement
has only ONE yelement associated
with it. 
The relation shown above is NOT a
function because the xelement 2 is paired with a
yelement of 4 and also a yelement of 6.
Similarly, the xelement of 1 is paired with the
yelements of 2 and 3.
The relation above can be altered to
become a function by removing the ordered pairs where
the xcoordinate 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 yvalues assigned to the same xvalue,
such as {(2,4), (2,6)}.
A function may, however, have two xvalues
assigned to the same yvalue, 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.


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 xvalue 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 yvalue in a
function,
especially when graphing. The yaxis if even
labeled as the f (x) axis, when graphing.
