
We know 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.
Given any x, there is only one y that can be paired with that x.
The following 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 set B 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.
OnetoOne Functions

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 value, there is only one x that can be paired with the given y.
Such functions are also referred to as injective.
When working on the coordinate plane, a function is a onetoone function when it will pass the vertical line test (to make it a function) and also a horizontal line test (to make it onetoone). 
Function f:
OnetoOne
Each y value that is used, is used only once.

Function g:
NOT OnetoOne
The yvalue of 4 is used more than once.

When working in the coordinate plane, the sets A (the domain) and B (the range)
will most often both become the Real Numbers,
stated as .
EXAMPLE 1: Is f (x) = x³ a onetoone function where ? 


This function is OnetoOne. 
This cubic function is indeed a "function" as it passes the vertical line test. In addition, this 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 a 11 function.
Notice that this function passes BOTH a vertical line test and a horizontal line test.

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

This absolute value function passes the vertical line test to be a function. In addition, this function has yvalues that are paired with more than one xvalue, such as (4, 2) and (0, 2). This function is not onetoone.
This function passes a vertical line test
but not a horizontal line test. 

EXAMPLE 3: Is g (x) =  x  2  a onetoone function where ? 
This question is changing the RANGE, not the DOMAIN.
It may be possible to adjust a function in some manner so that the function becomes a onetoone function. In this case, with set B, the range, redefined to be , function g (x) will still be NOT onetoone since we still have (0,2) and (4,2).
There are restrictions on the DOMAIN that will create a onetoone function in this example. For example, restricting A, the domain, to be only values from ∞ to 2 would work, or restricting A, the domain, to be only elements from 2 to ∞ would work. Notice that restriction A, the domain, to be would NOT create a onetoone function as we would still have (0,2) and (4,2). 
Onto Functions

A function f from A to B is called onto if for all b in B there is an a in A such that whenever f (a) = b. All elements in B are used.


Keep in mind that in an onto function, all possible yvalues are used.
Such functions are also referred to as surjective.
Function f:
Onto
All elements in B are used.
Not onetoone.

Function f:
NOT Onto
The 6 in B is not used.
It is onetoone. 
To determine if a function is onto, you need to know information about both set A and set B.
EXAMPLE 1: Is f (x) = 3x  4 is an onto function 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 function is also onetoone.

EXAMPLE 2: Is g (x) = x²  2 an onto function where ? 
This function
(a parabola)
is NOT ONTO. 

Values less than 2 on the yaxis are never used. Since only certain yvalues belonging to the set of ALL Real numbers are used, we see that not ALL possible yvalues are used.
Note that 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 an onto function where ?
If set B, the range, is redefined to be , ALL of the possible yvalues are now used, and function g (x) under these conditions) is ONTO. Note that this function is still NOT onetoone.
To make this function both onto and onetoone, we would also need to restrict A, the domain. 
BOTH
11 &
Onto Functions

A function f from A (the domain) to B (the range) is BOTH onetoone 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 onetoone and onto are referred to as bijective.
Bijections are functions that are both injective and surjective.
Function f:
BOTH
Onetoone and
Onto
Each used element of B is used only once, and All elements in B are used.

Function f:
NOT BOTH
Onetoone, NOT onto
Each used element of B is used only once, but the 6 in B is not used. 
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