Q. 185.0( 2 Votes )

# Give examples of two surjective function f_{1} and f_{2} from Z to Z such that f_{1} + f_{2} is not surjective.

Answer :

**TIP:** –

__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

Let, f_{1}: Z → Z and f_{2}: Z → Z be two functions given by (Examples)

f_{1}(x) = x

f_{1}(x) = – x

From above function it is clear that both are Onto or Surjective functions

Now,

f_{1} + f_{2} : Z → Z

⇒ (f_{1} + f_{2})(x) = f_{1}(x) + f_{2}(x)

⇒ (f_{1} + f_{2})(x) = x – x

⇒ (f_{1} + f_{2})(x) = 0

Therefore,

f_{1} + f_{2} : Z → Z is a function given by

(f_{1} + f_{2})(x) = 0

Since f_{1} + f_{2} is a constant function,

Hence it is not an Onto/Surjective function.

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