Functions - Domain, Range

Functions
Definition of a function :
    Definition : Let A and B be non empty sets and 'f' be a relation from A to B if for each element \( a \in A \) , there  exists a unique element \( b \in B \) such that \( \left( {a,b} \right) \in f \) then 'f' is called a function ormapping from A to B (or A into B) it is denoted by \( f:A \to B \)
Note : A is called domain and B is called co-domain of 'f'
    A function 'f' can also be seen in the following way
 
Note : A relation 'f' from A to B \( \left( {i.e\,\,\,\,f \subseteq A \times B} \right) \) is a function from A to B if for each \( a \in A \) there exists one \( b \in B \) such that \( \left( {a,b} \right) \in f \) and 'b' will be denoted by f(a) in other words \( (f,f(a)) \in f \).
 i.e f(a) = b

Range (definifion) If \( f:A \to B \) is a function, then f(A), the set of all f - images of elements of A, is called the range of 'f'
Clearly \( f(A) = \{ f(a)/a \in A\} \subseteq B \) and also \( f(A) = \{ b \in B/b = f(a)\} \)
Ex : i) Let \( f:N \to N \) be defined by f(n) =2n
         Range of f = \( f(N) = \{ 2n/n \in N\} \)
      ii) Let be defined by f(x) =x²
          range of f =\( f(R) = \{ x^2 /x \in R\} = [0,\infty )\,\,\left( {\because \,\,x^2 \geqslant 0} \right) \)
    iii) If \( f:A \to B \) and n(A)=m, n(B)=n, then the number of functions from A to B is n^m
Proof :  A = {x₁, x₂.....x_m}, B = {y₁,y₂,y₃......y_n}(\( \ne \)0)

x_m takes n images ,     
x₂ takes ‘n’ images 
............................
x_n takes ‘n’ images
total number of functions from A to B = n x n x .....(m itmes)
Total number of function = n^m