Saturday, September 2, 2023

function | domain and range

function

Function

Suppose that A and B be any two nonempty set. A operation  f : A → B is called a function from set A to set B if
  • all the element of set A is related to the elements of set B.
  • its mandatory that all the elements of set A should related with elements of another set.
  • each of the elements of set A is uniquely mapped with the elements of set B.
for example let's take  A={a,b,c,d} and B={1,2,3,4} then 


  • In given figure f1 is a function according to the definition as a maps to 2,b maps to 1,c maps to 4 and d maps to 3.
  • f2 is also a function according to definition as a maps to 1,b maps to 2,c maps to 3 and d maps to 4.
  • f3 is also a function and each elements of set A is uniquely mapped with elements 1 of B.
  • f4 is not a function as the element a of set A is not uniquely mapped with the elements of set B.
  • f5 is also not a function because the element d of set A does not mapped with any elements of set B.

Domain, Co-Domain and Range

Suppose  f : A → B be any function then Set A is called as the Domain of the function f, Set B is called as Co-Domain of function f. Set of elements of B which is the pre-image of set A under function f is called as the Range of function f ; i.e. Set of f(A) is the Range of the function f.

Problem
Consider A={-1,0,1} and B={0,1,2,3,4} and suppose that a function f(x)=x^2. then,
f(-1)=(-1)^2=1, f(0)=(0)^2=0 and f(1)=(1)^2=1
we can see here that each elements of Set A is uniquely mapped with Set B. hence f : A → B defined by f(x)=x^2 is a function.
Clearly the domain of function is {-1,0,1}, co-domain of function is {0,1,2,3,4} and the range of the function is {0,1}.

Equality of two function

two function f and h are said to be equal if and only if
  • domain of h = domain of f
  • co-domain of h = co-domain of f
  • h(x) = f(x) for all x belongs to their respective domain.
For Example Let   f : R→ R and g : R→ R be such that f(x) = x and g(x) = x^2/x for all x belongs to R(set of Real No.) 
Then we see that both function have the same domain and codomain in Set of Real Numbers and also both f(x) = g(x) for all x belong to respective domain of respective function; i.e. Set of Real Number.

Number of function

let A and B be two non-empty set having m and n elements respectively then the total number of function from Set A to Set B is equal n*n*n*............*n (m-times) that is equal to n^m.

Total number of Relation from Set A to Set B = 2^mn
Total number of Relation that is not function = 2^mn - n^m

One-One Function (Injective Function)

Let  f : A → B be any function then the function f is called one-one 
⟺f(a) = f(b) implies a=b for all a∈A and b∈b
⟺a ≠ b implies f(a) ≠ f(b)

one-one function
injective function

Example:  Let A={1,2,3} and B={1,2,3,4,5} and let  f : A → B be any function defined by f(x)=x+1 for all x∈A then we can observe that f ={(1,2),(2,3),(3,4)}.
Its clear that all the different elements of A have different image under the given function. hence, function f is an injective/one-one function.

Number of One-One Function

If A and B be any two non-empty set having m and n element respectively.Then the number of one-one function from Set A to Set B =nPm if n≥m And 0 if n<m
where nPm is permutational format where nPm=n!/(n-m)!

Note
  1. If f : A → B is an one-one function then the graph of the given function i.e.y = f(x) will be strictly decreasing or strictly increasing; i.e. dy/dx<0 or dy/dx>0
  2. If we draw a line parallel to X-Axis on the given function's Curve if there will only one intersecting point of the line on the given curve the function will be one-one but if the line cut the graph on distinct point function will not be injective.
Ex:
Let f : R→ R be any function such that f(x) = 2^x As the graph of given function is strictly increasing so by the first result we can conclude that f(x) is one-one/injective function.

increasing function
one-one function

Let f : R→ R be any function such that f(x)= X^2 then we see that by the figure if we draw a line parallel to X -Axis there will be two intersecting point so the function will not be one-one function.

Onto Function (Surjective Function)

Let  f : A → B be any function then the function f is called onto If every elements of Set B is the pre-image of some element of set A under the function f; i.e. B = f(A) 

onto function
onto and not onto function


Number of Onto Function
If Set A has m elements and Set B has n elements, the number of onto functions are given by the following formula
n^m-[nC1](n-1)^m+[nC2](n-2)^m........(-1)^n-1[nCn-1]1^m   

Eg. Let A={-3,-1,1,3} and B ={1,9} and Let's define a function f : R → R such that f(x)=X^2 then f(A)={f(-3),f(-1),f(1),f(3)}={1,9} so the function will be Onto.

Bijection Function(One-One Onto Function)

Let  f : A → B be any function then the function f is called Bijective function/One-One Onto function if It is Both One-One and Onto function. 

Number of Bijective Mapping 

Let  f : A → B be any bijective function then the Cardinality of both the set A and set B will same. Let's take Cardinality is m then the no. of bijection will be m! 

Constant Function 

Let  f : A → B be any function then the function f is called constant function if f(x)=c where c is any constant number and c∈B for all x∈A.

constant function
Note
  • Constant Function may be one-one or many-one, onto or into
  • Range of constant function contains singleton set or only one element.

Identity Function

Let  f : A → B be any function then the function f is called Identity function ⟺ f(x) = x for all x∈A. Identity function is denoted by I. Identity function is a Bijective function too.

identity function
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