Q. 1 A3.8( 17 Votes )

# Give an example of a function

Which is one – one but not onto.

Answer :

**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) = x^{2}

__Check for Injectivity__:

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

So, from definition

⇒ f(x) = f(y)

⇒ x^{2} = y^{2}

⇒ x^{2} – y^{2} = 0

⇒ (x – y)(x + y) = 0

As therefore x + y>0

⇒ x – y = 0

⇒ x = y

Hence f is One – One function

__Check for Surjectivity__:

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

⇒ f(x) = y

⇒ x^{2} = y

⇒

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

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

Hence, given by f(x) = x^{2} is One – One but not onto.

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