We know that a function is a set of ordered pairs such that given any x value,
there is only one y value that can be paired with that x value.

The following Venn diagrams depict functions:


Function f:
(m,3), (a,2), (t,9), (h,4)

Function g:
(m,3), (a,4), (t,9), (h,4)

In these diagrams, set A is the domain of the function, and
the elements from set B that are used is the range of the function.

With the definition of a function in mind, let's take a look at some special "types" of functions.


One to One
A function  f  from A to B is called one-to-one (or 1-1, or 1:1)
if whenever f (a) = f (b) then a = b.
And conversely, if  a ≠ b, then  f (a) ≠ f (b).
No element of B is the image of more than one element in A.
Such a function is also referred to as an injective function.

A one-to-one function is a function where every unique input maps to a strictly unique output.
In a one-to-one function, given any y value, there is only one x paired with that y value.

You can think of 1-1 functions as the input value being "married" to an output value. The input can only be legally "married" to one output value at a time, and that output value cannot be legally "married" to any other input value.


Function f:
One-to-One
Each y value that is used,
is used only once.

Function g:
NOT One-to-One
The y-value of 4 is
used more than once.



  I M P O R T A N T !!
In the coordinate plane, it is important that you remember two important "tests"
when working with one-to-one functions:

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

statement
1-1 function
Horizontal line test determines a one-to-one function:
 Any horizontal line intersects the graph of a "one-to-one function" in only ONE point.
A horizontal line represents a constant y-value. If it intersects the graph more than once,
the graph has used that y-value more than once, making it not a one-to-one function.

bullet When working on the coordinate plane, a function is a one-to-one function
when it will pass the vertical line test (to make it a function)
and also pass a horizontal line test (to make it a one-to-one function).

EXAMPLE 1:  Is f (x) = x³  a one-to-one function where fff?
ff

This function is One-to-One.

This cubic function is indeed a "function" as it passes the vertical line test. In addition, this 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 a 1-1 function.

Notice that this function passes BOTH the vertical line test and a horizontal line test, where any horizontal line intersects the graph only once.



EXAMPLE 2:  Is g (x) = | x - 2 | a one-to-one function where gg?

This function is
NOT One-to-One.

 g

This absolute value function passes the vertical line test to be a function. In addition, this function has y-values that are paired with more than
one x-value, such as (-4, 6) and (8, 6).
This function is not one-to-one.

This function passes a vertical line test
but not a horizontal line test.



EXAMPLE 3:  Is g(x) = | x - 2 |, as seen in the graph for Example 2, a one-to-one function when the mapping is limited such that ffs?

beware This question is changing the RANGE, not the DOMAIN.

It may be possible to adjust (restrict) a function in some manner so that the function becomes a one-to-one function. In this case, with set range, redefined to be ffss, function g(x) will still NOT be one-to-one since we still have (-4, 6) and (8, 6) failing the horizontal line test.

There are restrictions on the DOMAIN that will create a one-to-one function in this example, such as:
• restrict the domain, to be only values from -∞ to 2.
• restrict the domain, to be only elements from 2 to ∞.
Notice that restricting the domain, to be ffss would NOT create a one-to-one function as we would still have (-4,6) and (8,6) that fail the horizontal line test.


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There is another type of function often seen when working with 1-1 functions.


Onto
An onto function is a mathematical function where every possible output is mapped to by at least one input element of the domain.. All available output elements are used.
Such functions are also referred to as an surjective function.

Keep in mind that in an onto function, all possible y-values are used.



Function g:
Onto
All elements in B are used.
Not one-to-one,
but it is onto.

Function f:
NOT Onto
The 6 in B is not used.
It is one-to-one,
but it is not onto..

To determine if a function is onto, you need to know information about both set A and set B.



In the coordinate plane, onto functions must pass the vertical lines test (since they are functions). These functions, however, are not necessarily one-to-one functions
(so may or may not pass the horizontal line test).

EXAMPLE 1:  Is f (x) = 3x - 4 an onto function where hhhh?
hhhhh
 
h

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 function is both one-to-one
and onto. 



EXAMPLE 2:  Is g (x) = x² - 2 an onto function where hhh?
 

This function
(a parabola)
is NOT ONTO.

 hh
Values less than -2 on the y-axis are never used. Since only certain y-values belonging to the set of ALL Real numbers are used, we see that not ALL possible y-values are used.
This function is not an onto function.

In addition, this parabola has y-values that are paired with more than one x-value,
such as (3, 7) and (-3, 7).
This function is not one-to-one.
hhhhhh




EXAMPLE 3:  Is g (x) = x² - 2, as graphed in Example 2, an onto function where fd?
If set the range is redefined to be fdd, ALL of the possible y-values are now used in this restricted setting, and  function g (x) can now be considered as ONTO.
This function, however, is still NOT one-to-one.

To make this function both onto and one-to-one, we would also need to restrict the domain to ensure that the horizontal line test would intersect only once..


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There are functions that are both 1:1 and Onto functions.

BOTH
1-1 & Onto Functions
A function f from A (the domain) to B (the range) is BOTH one-to-one and onto when no element of B is the image of more than one element in A, AND all elements in B are used.
Functions that are both one-to-one and onto are referred to as bijective.
Bijections are functions that are both injective and surjective.



Function f:
One-to-one, NOT onto
Each element of B (that is used) is used only once, but not ALL elements of B are used
(the 6 in B is not used).
Amended Function f:
BOTH
One-to-one and Onto
Each element of B is used only once, and ALL elements of B are used.

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