
We saw that a relation is simply a set of input and output values, represented in ordered pairs.
No special rules applied to a relation.
If we add a "specific rule" to a relation, we get a function.
A function is a relation with a specific rule.
The function "rule" is that a function has only one relationship for each input value.
Remember, under the definition of a relation we saw that
an xvalue could have more than one yvalue.
Relation: {(1,1), (2,3),(5,5),(5,6)}
(The x = 5 has two possible yvalues (both 5 and 6).)
This is not possible in a function.

A function is a set of ordered pairs in which each xelement has only ONE yelement associated with it.



A function is like a factory machine that has input and output.
The input is called the "domain",
and the output is called the "range"
of the function.
Each value "enters" the function only once, and has only "one value coming out" of the function.

While a function may NOT have two yvalues assigned to the same xvalue,
it may have
two xvalues assigned to the same yvalue.
NOT OK for a function:
{(5,1),(5,4)} 
OK for a function:
{(5,2),(4,2)} 
Function: each xvalue has only ONE yvalue!


Let's adjust our previous "relation" example so it fits the function "definition".
The relationship between eye color and student names.
(x,y) = (eye color, student's name)
Relation: Set A = {(green,Steve), (blue,Elaine), (brown,Kyle), (green,Marsha), (blue,Miranda), (brown, Dylan)}


If we remove duplicate eye colors,
this eye color example will be a function:
Function: Set B =
{(blue,Steve), (green,Elaine), (brown,Kyle)}

Let's adjust this previous "relation" example so it fits the function "definition".
Relation:
(where xvalues have
more than one yvalue)
Relation:
{(1,1),(1,2),(3,3),(4,4),(5,5),(5,6),(6,4)}

Function:
If we remove (1,2) and (5,6),
we can create a function.
Relation:
{(1,1),(3,3),(4,4),(5,5),(6,4)}

Let's adjust this previous "relation" example so it fits the function "definition".
Notice that vertical lines on the graphs make it clear if an xvalue had more than one yvalue.
If the vertical lines intersected the graph in more than one location, we had a relation, NOT a function.

Vertical line test for functions: Any vertical line intersects the graph of a function in only ONE point. 

Given that relation A = {(4,3), (k,5), (7,3), (3,2)}. Which of the following values for k will make relation A a function? Choices: a) 3 b) 4 c) 6
Solution: Choice c. The xvalues of 3 and 4 are already used in relation A. If they are used again (with a different yvalue), relation A will not be a function, as those xvalues will be used more than once.
Which of the following graphs represents a function?
Solution: Choice b. A vertical line drawn on this graph will intersect the graph in only one location, making it a function. Vertical lines on the other three graphs will intersect the graph in more than one location, or as in part a, will intersect in an infinite number of points (all points).
Function Notation:
(you may, or may not, be using function notation) 
Function notation is the way a function is written. It is meant to be a precise way of giving information about the function without a rather lengthy written explanation.
The most popular function notation is f (x) which is read "f of x".
This is NOT the multiplication of f times x.. 


Traditionally, functions are referred to by single letter names, such as f, g, h and so on.
Any letter(s), however, may be used to name a function.
Examples:
The f (x) notation is another way of representing the yvalue in a function, y = f (x).
The yaxis may even be labeled as the f (x) axis, when graphing.
Ordered pairs may be written as (x, f (x)), instead of (x, y).
Note: The notation f : X → Y tells us that the function's name is "f " and its ordered pairs are formed by an element x from the set X, and by an element y from the set Y.
( The arrow → is read "is mapped to". )


Calculators graph functions! 

Most calculators (including the TI84+ series) can only handle graphing functions.
The equation (function) must be in "y = " form before you can enter it in the calculator.
By solving for "y =", you are actually identifying a "function".
BUT ... what about y^{2} = x ?
If we solve for "y =", we get , which (seen in Example 3) is not a function.
This equation cannot be solved for a unique (only one) "y =" equation.
We cannot graph this on our calculator as a single entry, since there is no key for "±".
We actually have two "y =" equations: and .
(Yes, the graphing calculator can graph these equations separately to form the graph.
But the combined graphs will be a relation, not a function.)
The lack of a unique (one) "y =" equation means that y^{2} = x is not a function.
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