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 x-value could have more than one y-value.
Relation: {(1,1), (2,3),(5,5),(5,6)}
(The x = 5 has two possible y-values (both 5 and 6).)
This is not possible in a function.

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

domainfactory

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 y-values assigned to the same x-value,
it may have two x-values assigned to the same y-value.

NOT OK for a function:
{(5,1),(5,4)}
OK for a function:
{(5,2),(4,2)}

Function: each x-value has only ONE y-value!
functionstr

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ex1

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)}

eye color
If we remove duplicate eye colors, this eye color example will be a function:

Function: Set B = {(blue,Steve), (green,Elaine), (brown,Kyle)}



ex2

Let's adjust this previous "relation" example so it fits the function "definition".

Relation:
(where x-values have
more than one y-value)
relation1
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.
relation4
Relation:
{(1,1),(3,3),(4,4),(5,5),(6,4)}


ex3

Let's adjust this previous "relation" example so it fits the function "definition".

Relation:
(where x-values have
more than one y-value)
relation2
Relation: relationmath1; allows for points
such as (
2,1.424) and (2,-1.414),
also (
4, 2) and (4,-2) .

Function:
If we change the ± sign to just a + sign,
we can create a function.
relation2
Function:  
relationmath1 where
each x-value has only ONE y-value.
(assuming x is 0 or larger)

 

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Notice that vertical lines on the graphs make it clear if an x-value had more than one y-value.
If the vertical lines intersected the graph in more than one location, we had a relation, NOT a function.

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

 

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ex4

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 x-values of 3 and 4 are already used in relation A. If they are used again (with a different y-value), relation A will not be a function, as those x-values will be used more than once.

ex5

Which of the following graphs represents a function?

simple4
a)
simple2
b)
simple3
c)
simple4
d)

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).

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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..
functionnotpic

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:
domainmath11domainmath11b

The f (x) notation is another way of representing the y-value in a function, y = f (x).
The y-axis 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". )


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statement
Calculators graph functions!

Most calculators (including the TI-84+ 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 y2 = x ?
If we solve for "y =", we get relationmath1, 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: relationmath1 and math3.
(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 y2 = x is not a function.


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