Let A = [–1, 1], Then, discuss whether the following functions from A to itself are one – one, onto or bijective:

h(x) = x²

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, here f : A → A : A = [–1, 1] given by function is h(x) = x²

Check for Injectivity:

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

⇒ h(x) = h(y)

⇒ x² = y²

⇒ ±x = ±y

Since it has many elements of A co – domain

Hence, h is not One – One function

Check for Surjectivity:

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

⇒ h(x) = y

⇒ x² = y

⇒ x = ±√y

Since h have no pre–image in domain A.

Hence, h is not onto function

Thus, It is not Bijective function