Answer :

**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, Let, given by f(x) = x^{3} + x

__Check for Injectivity__:

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

So, from definition

⇒ f(x) = f(y)

⇒ x^{3} – x = y^{3} – y

⇒ x^{3} – y^{3} – (x – y) = 0

⇒ (x – y)(x^{2} + xy + y^{2} – 1) = 0

Hence f is not One – One function

__Check for Surjectivity__:

Let y be element belongs to R i.e be arbitrary, then

⇒ f(x) = y

⇒ x^{3} – x = y

⇒ x^{3} – x – y = 0

Now, we know that for 3 degree equation has a real root

So, let be that root

⇒

Thus for clearly , there exist such that f(x) = y

Therefore f is onto

Thus, It is not Bijective function

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