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, given by f(x) = 1 + x²

Check for Injectivity:

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

So, from definition

⇒ f(x) = f(y)

⇒ x² + 1 = y² + 1

⇒ x² = y²

⇒ ±x = ±y

Therefore, either x = y or x = – y or x ≠ y

Hence f is not One – One function

Check for Surjectivity:

1 be element belongs to R i.e be arbitrary, then

⇒ f(x) = 1

⇒ x² + x = 1

⇒ x² + x – 1 = 0

⇒

Above value of x not belongs to R for y < 1

Therefore f is not onto

Thus, It is also not Bijective function