Properties of functions

[Sejal Anand]

Even after researching a lot, I still face problem in identifying one-one(injective) , onto(surjective) and bijective functions.

Can anyone please suggest me how to work on problems based on these.

[Debajyoti Biswas]

Bijective, surely you know, happens when a function is both injective and surjective. 

 For injective function, every element of the function's codomain is the image of at most one element of its domain.  (there is an individual relation between the elements in domain and codomain) 

Now, for surjective,  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.

ØFor functions that are given by some formula there is a basic idea. We use the definition of injectivity, which is if f(x) = f(y), then x = y

•Example:

•f(x)=2x+3. Now, assume two var. m and n, such that 2m+3=2n+3,  this means  m=n, so this function is injective

•The function f(x) = 2x + 1 over the  real number set (f: ℝ -> ℝ ) is surjective because for any real number y, you can always find an x that makes f(x) = y true; in fact, this x will always be (y-1)/2 .

The quadratic function f(x) = x2 is not a surjection. There is no x such that x2 = −1. The range of x² is [0,+∞) , that is, the set of non-negative numbers. (Also, this function is not an injection.)