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