RelationsA Relation is a rule that links
one or more output to one or more input
An input is anelement in the starter
set. It can also be called x
value
An output is anelement in the arrival
set.It can also be called y
value
Inputs Outputs
FunctionsA relation is a Function if all inputs
are mapped to only one output
This is a function
All inputs are mapped
All inputs have only one output
All inputs are mapped
All inputs have only one output
This is not a function
For example…
State Capital
New Jersey Trenton
New York Albany
Colorado Denver
Rhode Island Providence
Person Grandparent
Eric Arthur
Eric Ellie
Maria Camine
Maria Susan
This tab, where each state ismapped with its capital, represents a function, becauseevery state (inputs) has only onecapital (outputs).
This tab instead, that maps everyperson with his grandparents, isn’t a function, because they both havetwo grandparents (outputs).
x y
5 7
2 9
0 8
-4 6
x y
4 3
2 8
0 8
4 6
Here, each number in the x (inputs) column has only oney (output) as an answer, so itis a function.
Here, the input 4 has anoutput of both 3 and 6, so itcan’t be a function.
How to represent a function
A function can be represented in three ways:1-the graph2-the arrow chart3-the set of ordered pairs
The graph
The graph of a function is a cartesian coordinate system where the couples of x and y values are represented.
this is an exampleof the graph of the function:y=2x+6
The arrow chart
The arrow chart of a function is another graphical representation
The arrows show how the numbers in the domain are mapped into numbers in the range and represent the function f: y=f(x)
The set of ordered pairs
a set of ordered pairs is a non-graphical way to represent a function;the couples of x and y values will be represented as shown in the followingway:
y=2x+6----> {(1,8),(2,10),(3,12)}
X Y
1 3
2 10
3 12
1 Is the following a function?
Yes
No
1
2
3
4
5
4
3
2
A
B
1 Is the following a function?
1
2
3
4
5
4
3
2
Because each input is mapped into only one output
Correct!!
1
1
2
3
4
5
4
3
2
Is the following a function?
Wrong!It is a function because each input is mapped into only one output
2 Is the following a function?
City State
Englewood New Jersey
Englewood Colorado
Springfield Maryland
Springfield Illinois
Springfield Montana
Yes
No
A
B
2 Is the following a function?
City State
Englewood New Jersey
Englewood Colorado
Springfield Maryland
Springfield Illinois
Springfield Montana
Correct!!!
2 Is the following a function?
City State
Englewood New Jersey
Englewood Colorado
Springfield Maryland
Springfield Illinois
Springfield Montana
Wrong!
Vertical Line Test: if every vertical line you can draw goes through only 1 point then the relation is a function.
Function Function Not a function
Example
Not a function
This isn’t a functionbecausethe vertical line goesthroughtwo points.
Is this a function?
Yes
No
A
B
Is this a function?
Correct!!!
Is this a function?
Wrong!
Is this a function?
Yes
No
A
B
Is this a function?
Correct!!!
Is this a function?
Wrong!
Is this a function?
Yes
No
A
B
Is this a function?
Correct!!!
Is this a function?
Wrong!
Domain codomain and range
Any set of numbers that is the input of a relation or function is called the domain.
Any set of numbers that is the output of a relation or function is called the range.
Domain: {1, 2, 3, 4} Range: {42, 98, 106, 125}
f(x) Y=2x+1
32
DOMAIN: a set of first elementin a relation ( all of the x values ). These are also called the independent variable.
RANGE:the second and actualselements in a relation ( all of the y values ). These are also calledthe dependent variable
1
3
4
5
7
9
CODOMAIN: is the set of all possible and second elements in a relation.
• Domain = {Joe, Mike, Rose, Kiki}
• Range = {6, 5.75, 5, 6.5}
Example
Joe
Mike
Rose
kiki
DOMAIN RANGE
6
5,75
5
6,75
Domain (set of all x’s) Range (set of all y’s)
1
2
3
4
5
2
10
8
6
4
Which numbers are in the domain?
2
4
6
8
10
10
20
30
40
50
2
10
12
8
20
A
B
C
D
E
Which numbers are in the domain?
2
4
6
8
10
10
20
30
40
50
2
10
8
A
B
D
Which numbers are in the range?
2
4
6
8
10
10
20
30
40
50
2
10
12
8
20
A
B
C
D
E
Which numbers are in the range?
2
4
6
8
10
10
20
30
40
50
10
20
B
E
Which numbers are part of the domain?
{(24, 12), (22, 11), (20, 10), (18, 9)}
12
20
24
11
9
A
B
C
D
E
Which numbers are part of the domain?
{(24, 12), (22, 11), (20, 10), (18, 9)}
20
24
B
C
Which numbers are part of the range?
{(24, 12), (22, 11), (20, 10), (18, 9)}
12
20
24
11
9
A
B
C
D
E
Which numbers are part of the range?
{(24, 12), (22, 11), (20, 10), (18, 9)}
12
11
9
A
D
E
INJECTIVE, SURJECTIVE and BIJECTIVE FUNCTIONS.
•
•
•
• •
•
• •
X
Y
Let f be a function whose domain is a set X. The function f is injective if for all a and b in X, if f(a) = f(b) then a = b; Equivalently, if a ≠ b, then f(a) ≠ f(b).So….F(x) is injective (also called one-to-one) if no twoinputs have the same output.
This is an INJECTIVE function.
This isn’t an INJECTIVE function because we can
see that two differentelements ‘3;4’ in the
domain have the sameoutput ‘C’.
To be a one-to-one function, each y value could only be
paired with one x. Let’s look at a couple of graphs.
Look at a y value (for example
y = 3) and see if there is only
one x value on the graph for it.
This is NOT a one-to-one
function
For any y value, a horizontal line
will only intersect the graph once
so will only have one x value
This then IS a one-to-one
function
If a horizontal line intersects the graph of an
equation more than one time, the equation
graphed is NOT a one-to-one function.
This is an injective
funtion.
This is NOT an
injective funtion.
This is NOT an
injective funtion.
•
•
• •
•
•
•
X
Y
A function f (from set X to Y) is surjective if and only if forevery b in Y, there is at least one a in X.So…A function is surjective (onto) if every element in the codomain (Y) has some element from the domain (X) mappedon to it.In other words: f(x) is surjective when the range is equal to the codomain.
It’s surjective because allelements in the codomain
has an input
It isn’t surjective becausethere is an element in the codomain Y that hasn’t an
input
•
•
•
•
•
•
A
B
A function f (from set A to B) is bijective if, for every y in B, there is exactly one x in A such that f(x) = y .So…A function is bijective if it’s both injective and surjective.
•Is injective because each input has a different output.
•is surjective because the range is allcodomain.
It’s BIJECTIVE
a
b
c
1
2
This function is:•injective•surjective•bijective•neither
a
b
c
1
2
3
This function is:•injective•surjective•bijective•neither
Davide
Lia
Ugo
Irene
Marco
Erika
This function is:•injective•surjective•bijective•neither
Tom
Gigi
Carla
Sammy
Anna
This function is:•injective•surjective•bijective•neither
A: {Maria; Francesca; Vanessa; Gaia}B: {Edoardo; Andrea; Luca; Davide}
A B
Maria Andrea
Vanessa Edoardo
Francesca Luca
Gaia Davide
This is a function:•Injective•Surjective•Bijective
A: {Maria; Vanessa; Gaia}B: {Edoardo; Andrea; Luca; Davide}
A B
Maria Andrea
Vanessa Edoardo
Gaia Davide
This is a function:•Injective•Surjective•Bijective
A: {Maria; Francesca; Vanessa; Gaia; Ilaria}B: {Edoardo; Andrea; Luca; Davide}
A B
Maria Andrea
Vanessa Edoardo
Francesca Luca
Gaia Edoardo
Ilaria Davide
This is a function:•Injective•Surjective•Bijective
Top Related