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

Now, Let, given by f(x) = x³ – x

Check for Injectivity:

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

So, from definition

⇒ f(x) = f(y)

⇒ x³ – x = y³ – y

⇒ x³ – y³ – (x – y) = 0

⇒ (x – y)(x² + xy + y² – 1) = 0

As x² + xy + y² ≥ 0

⇒ therefore x² + xy + y² – 1≥ – 1

⇒ x – y≠0

⇒ x ≠ y for some

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

⇒ x³ – x = y

⇒ x³ – 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

⇒ Hence, given by f(x) = x³ – x is not One – One but onto