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

So, f: A → B is One – One function

⇔ a≠b

⇒ f(a)≠f(b) for all a, b ∈ A

⇔ f(a) = f(b)

⇒ a = b for all a, b ∈ A

Onto Function: – A function f: A → B 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, f: A → B is Surjection iff for each b ∈ B, there exists a ∈ B such that f(a) = b

Now, Let, f: R → R given by f(x) = x³ – x

Check for Injectivity:

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

Hence f is not One – One function

Check for Surjectivity:

Let y be element belongs to R i.e y ∈ R 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 x = α be that root

⇒ α³ - α = y

f(α) = y

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

Therefore f is onto

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