Types of Functions

A funcion is a connection between 2 sets A and B f: A→B such that
1) All elements in A are associated to some element in B
2) This association is unique, that means one and only one.
Let's try to understand this with a simple anology,

Here, let's say `"X"_1` is a set of all children and `"X"_2` is a set of all womens. And `"X"_1` and `"X"_2` have connection as mother and children. 
So as per the definition there is a connention between 2 sets `"X"_1` and `"X"_2` such that all the elements of `"X"_1` are associated to some element in set `"X"_2` i.e all the childrens are related to a particualr mother, and this association is unique because no one child can have two or more mothers, but a mother can have more than one child. 
Examples-
X= {1,2,3,4}
Y= {1,5,,9,11,15,16}
Are the following relations functions?
a) `"f"_1`= {(1,1), (2,11), (3,1), (4,15)}
As per the definition this is the function.
b) `"f"_2`= {(1,1), (2,7), (3,5)}
This is not a function because 4 is not related with any element in set Y.
c) `"f"_3`= {(1,5), (2,9), (3,1), (4,5), (2,11)}
This is not a function because 2 is associated to 9&11.
This can also be written as `"f"_3`(2)= 9 and       `"f"_3`(2)= 11 
i.e f(x)=y
(x,y)∈f
⦁ Domain of a Function- Domain is known as the input of a functin where all the values of x, for which f(x) stays defined or gives a valid answer.
                Example- f(x)= x^3- 3
               x∈R, so R is the domain here.
⦁ Range of a Function- It the ouput of a function. The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain. The definition means the range is the resulting y-values we get after substituting all the possible x-values.
               Example- `f(x)= sqrt(x-1)`
               `y= sqrt(x-1)`
               `y^2= x-1`
               `x= y^2+ 1`
              y∈R, so R is the range here. 
⦁ Co-domain of a funcion- The codomain or target set of a function is the set Y into which all of the output of the function is constrained to fall. It is the set Y in the notation f: X → Y.
Types of functions-
1) One one (injective) function- If each element have single image, the it is a one one function.

 Example- f(x)= 2x+3
if x=1, then y=5
x=2, y=7

`x=1/2, y= 4`  Here, every element of x have a separate value of y, thus it is a one one function.
One one function is further split into two parts
i) Onto one one function-  If each element of the codomain is mapped to by at least one element of the domain is an onto one one function. Co-domain= Range i.e y= f(x)

 ii)Into one one function- If each element of the codomain is not mapped to any one element of  the domain it is an into one one funcion.

 2) Many one (surjective) function- In mathematics, a function f from a set X to a set Y is surjective, or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x) = y. It is not required that x be unique; the function f may map one or more elements of X to the same element of Y.

 Example- `"f"(x)= x^2`
if x=1, then y= 1
x= -1, y= 1
x=2, y= 4
x= -2, y= 4
Here, element of y have more than one values of x, thus it is a onto function. 
Many one function is further split into two parts
i) Onto many one funcion- If two or more element of domain have one image and codomain is equal to range then it is said to be Onto many one function. 

 
ii) Into many one function- If two or more element of domain have one image and codomain is not equal to range then it is said to be Into many one function.