Remember that a function is
a set of ordered pairs in which no two ordered pairs
that have the same first component have different second
components. This means that given any x,
there is only one y that can be paired with that
x.
A function f from A to
B is called onto
if for all b in B there is an a in
A such that f (a) = b.
All elements in B are used.
Such functions are referred to as
surjective.
"Onto"
(all elements in B are used) 
NOT "Onto"
(the 8 and 1 in Set B are not used) 
By definition, to determine if a function is ONTO,
you need to know information about both set A and B.
When working in the coordinate plane,
the sets A
and B
may both become the Real numbers,
stated as
.
EXAMPLE 1: Is
f (x) = 3x  4
onto where
? 


This
function (a
straight line) is
ONTO. 
As you progress
along the line,
every possible yvalue is used.
In addition,
this straight line also possesses the property
that each xvalue has one unique yvalue
that is not used by any other xelement.
This characteristic is referred to as being
onetoone. 
EXAMPLE 2: Is
g (x) = x²  2
onto where
? 
This
function (a
parabola) is
NOT
ONTO. 

Values less
than 2 on the yaxis are never used.
Since possible yvalues belong to the set
of ALL Real numbers, not ALL possible yvalues
are used.
In addition, this parabola
also has yvalues that are
paired with more than one xvalue, such
as (3, 7) and (3, 7).
This function will not be
onetoone.


EXAMPLE 3: Is
g (x) = x²  2
onto where
?
If set
B is
redefined to be
,
ALL of the possible yvalues are now
used, and function g (x)
(under these conditions) is
ONTO. 
A function f from A
to B is called
onetoone (or 11) if whenever
f (a) = f (b) then a = b.
No element of B is the image of more than one
element in A.
In a onetoone function, given
any y there is only one x that can be
paired with the given y. Such functions are
referred to as injective.
"OnetoOne" 
NOT "OnetoOne" 
EXAMPLE 1: Is
f (x) = x³ onetoone where
? 


This
function is
OnetoOne. 
This cubic
function possesses the property
that each xvalue has one unique yvalue
that is not used by any other xelement.
This characteristic is referred to as being
11.
Also, in this
function, as you progress
along the graph,
every possible yvalue is used, making
the function onto.

EXAMPLE 2: Is
g (x) =  x  2 
onetoone where
? 
This
function is
NOT
OnetoOne. 

This absolute
value function has yvalues that are
paired with more than one xvalue, such
as (4, 2) and (0, 2).
This function is not
onetoone.
In addition, values less
than 0 on the yaxis are never used,
making the function NOT
onto.


EXAMPLE 3: Is
g (x) =  x  2 
onetoone where
? 
With set
B redefined to be
,
function g (x) will still be
NOT onetoone,
but it will now be
ONTO. 
Functions can be
both onetoone and
onto.
Such functions are called
bijective.
Bijections are functions that are both injective
and surjective.
"Both" 
NOT "Both"  not Onto 
Examples of functions that are BOTH
onto and
onetoone
can be seen in each of the categories above.