
You have worked with relations and functions in the past.
Let's refresh our memories and add a few more details.


A relation is simply a set of input and output values, represented in ordered pairs.
It is a relationship between sets of information. 

A relation can be any set of ordered pairs.
No special rules need apply to a relation.
The following is an example of a relation:
Relation:
{(1,1),(1,2),(3,3),(4,4),(5,5),(5,6),(6,4)}
The scatter plot graph at the right shows that a vertical line may intersect more than on point in a relation.
Both the xvalue of 1, and the xvalue of 5,
have more than one corresponding yvalue. 


The relationship between eye color and student names.
(x,y) = (eye color, student's name)
Set A = {(green, Steve), (blue, Elaine), (brown, Kyle),
(green, Marsha), (blue, Miranda), (brown, Dylan)}
This is a relation.
Notice that the xvalues (eye colors) get repeated. 
The graph, shown at the right, is also an example of a relation.
Notice that the xvalues may have
more than one corresponding yvalue.
The vertical red line points out that
the xvalue of 2 has more than one
corresponding yvalue.
This is a relation.

Relation: ; allows for points
such
as
(2,1.414) and (2,1.414).

If you add a "specific rule" to a relation, you get a function.

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



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 examples so they fit the function "definition".
If we remove (1, 2) and (5, 6),
we will have a function.
Function:
{(1,1),(3,3),(4,4),(5,5),(6,4)}

If we change the ± sign to just a + sign,
we will have a function.
Function:

If we remove duplicate eye colors,
the eye color example will be a function:
(x,y) = (eye color, student's name)
Set B =
{(blue, Steve), (green, Elaine), (brown, Kyle)}
Set B is a function. 

I M P O R T A N T !!
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? a) 3 b) 4 c) 6
Solution: Choice c. The xvalues of 3 and 4 are already assigned a yvalue in relation A.
If they are used again (with a different yvalue), relation A will not be a function.
Which of the following graphs represents a function?
Solution: Choice b. A vertical line drawn anywhere 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).

Calculators graph functions! 

When you want to graph lines, you typically:
1. solve the equation for "y =", produce a chart of points (a Tchart), and plot, or
2. solve the equation for "y =" plot the yintercept and use slope to plot the line, or
3. solve the equation for "y ="
and enter the equation in your graphing calculator. 
By solving for "y =", you are actually identifying a "function".
If you can solve an equation for "y =", then the equation is a "function".
4x + 2y = 10
2y = 4x + 10
y = 2x + 5 (a function!)
This equation can now be entered into a graphing calculator for plotting.
Most calculators (including the TI84+ series) can only handle the graphing of functions.
The equation (function) must be in "y = " form before you can enter it into the calculator. 
BUT ... what about y^{2} = x ?
If we solve for "y =", we get , which we saw, at top of this page, was not a function.
We cannot graph this on our calculator as a single entity, since there is no key for "±".
We were not able to solve this equation for a unique (only one) "y =" equation.
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 "y =" equation means that y^{2} = x is not a function.
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