  Question

Give examples of two surjective function $${f}_{1}$$ and $${f}_{2}$$ from $$Z$$ to $$Z$$, such that $${f}_{1}+{f}_{2}$$ is not surjective.

Solution

A function $$f:A\rightarrow B$$ is said to be a onto function or surjective if every elemnt of $$A$$ i.e, if $$f(A)=B$$ or range of $$f$$ is the co-domain of $$f.$$So, $$f:A\rightarrow B$$ is surjective for each $$b\in B,$$ there exists $$a\in B$$ such that $$f(a)=b$$Let $$f_1:Z\rightarrow Z$$ and $$f_2:Z\rightarrow Z$$ be two functions given by$$\Rightarrow$$  $$f_1(x)=x$$$$\Rightarrow$$  $$f_2(x)=-x$$From above function it is clear that both are surjective functions.Now,$$\Rightarrow$$  $$f_1+f_2:Z\rightarrow Z$$$$\Rightarrow$$  $$(f_1+f_2)(x)=f_1(x)+f_2(x)$$$$\Rightarrow$$  $$(f_1+f_2)(x)=x-x$$$$\Rightarrow$$  $$(f_1+f_2)(x)=0$$Therefore, $$f_1+f_2:Z\rightarrow Z$$ is a function given by$$\Rightarrow$$  $$(f_1+f_2)(x)=0$$Since, $$f_1+f_2$$ is a constant function, hence it is not an onto or surjective function.Mathematics

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