TIP: – One – One Function: – A function is said to be a one – one functions or an injection if different elements of A have different images in B.

So, is One – One function

⇔ a≠b

⇒ f(a)≠f(b) for all

⇔ f(a) = f(b)

⇒ a = b for all

Onto Function: – A function is said to be a onto function or surjection if every element of A i.e, if f(A) = B or range of f is the co – domain of f.

So, is Surjection iff for each , there exists such that f(a) = b

Bijection Function: – A function is said to be a bijection function if it is one – one as well as onto function.

Now, f : Z → Z given by f(x) = x² + x

Check for Injectivity:

Let x,y be elements belongs to Z i.e such that

⇒ f(x) = f(y)

⇒ x² + x = y² + y

⇒ x² – y² + x – y = 0

⇒ (x – y)( x + y + 1) = 0

Either (x – y) = 0 or ( x + y + 1) = 0

Case i :

If x – y = 0

⇒ x = y

Hence f is One – One function

Case ii :

If x + y + 1 = 0

⇒ x + y = – 1

⇒ x ≠ y

Hence f is not One – One function

Thus from case i and case ii f is not One – One function

Check for Surjectivity:

As

Let x be element belongs to Z i.e be arbitrary, then

⇒ f(x) = 1

⇒ x² + x = 1

⇒ x² + x – 1 = 0

⇒

Above value of x does not belong to Z

Therefore no values of x in Z (co – domain) have a pre–image in domain Z.

Hence, f is not onto function

Thus, Not Bijective also