One-to-one and Onto Functions
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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.

Onto Function

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) = x - 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) = x - 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.
 

 

One-to-One Function

 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) = x  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.

 

BOTH

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.