Classify the following functions as injection, surjection or bijection:

f : Z → Z given by f(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, f : Z → Z given by f(x) = x²

Check for Injectivity:

Let x₁, – x₁ be elements belongs to Z i.e such that

So, from definition

⇒ x₁ ≠ – x₁

⇒ (x₁)² = ( – x₁)²

⇒ f(x₁)² = f( – x₁)²

Hence f is not One – One function

Check for Surjectivity:

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

⇒ f(x) = y

⇒ x² = y

⇒

⇒ not belongs to Z for non–perfect square value of y.

Therefore no non – perfect square value of y has a pre–image in domain Z.

Hence, f is not Onto function.

Thus, Not Bijective also