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 y-value is used.
In addition,
this straight line also possesses the property
that each x-value has one unique y-value
that is not used by any other x-element.
This characteristic is referred to as being
one-to-one. |
EXAMPLE 2: Is
g (x) = x² - 2
onto where
 ? |
|
This
function (a
parabola) is
NOT
ONTO. |
 |
Values less
than -2 on the y-axis are never used.
Since possible y-values belong to the set
of ALL Real numbers, not ALL possible y-values
are used.
In addition, this parabola
also has y-values that are
paired with more than one x-value, such
as (3, 7) and (-3, 7).
This function will not be
one-to-one.
|
|
EXAMPLE 3: Is
g (x) = x² - 2
onto where
 ?
If set
B is
redefined to be
 ,
ALL of the possible y-values are now
used, and function g (x)
(under these conditions) is
ONTO. |
A function f from A
to B is called
one-to-one (or 1-1) 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 one-to-one function, given
any y there is only one x that can be
paired with the given y. Such functions are
referred to as injective.
|
"One-to-One" |
NOT "One-to-One" |
|
EXAMPLE 1: Is
f (x) = x³ one-to-one where
? |
|
|
|
|
This
function is
One-to-One. |
This cubic
function possesses the property
that each x-value has one unique y-value
that is not used by any other x-element.
This characteristic is referred to as being
1-1.
Also, in this
function, as you progress
along the graph,
every possible y-value is used, making
the function onto.
|
EXAMPLE 2: Is
g (x) = | x - 2 |
one-to-one where
? |
|
This
function is
NOT
One-to-One. |
|
This absolute
value function has y-values that are
paired with more than one x-value, such
as (4, 2) and (0, 2).
This function is not
one-to-one.
In addition, values less
than 0 on the y-axis are never used,
making the function NOT
onto.
|
|
EXAMPLE 3: Is
g (x) = | x - 2 |
one-to-one where
 ? |
With set
B redefined to be
 ,
function g (x) will still be
NOT one-to-one,
but it will now be
ONTO. |
Functions can be
both one-to-one 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
one-to-one
can be seen in each of the categories above.
