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Check the injectivity and surjectivity of the following functions f defined from Z to Z given by f(x) = x to the power 2

Q.2 Check the injectivity and surjectivity of the following functions:

(ii) f : Z \rightarrow Z given by f(x) = x^2

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f : Z \rightarrow Z

f(x) = x^2

One- one:

For   -1,1 \in Z  then f(x) = x^2

                                  f(-1)= (-1)^{2}

                                  f(-1)= 1   but  -1 \neq 1 

\therefore  f is not one- one i.e. not injective.

For  -3 \in Z  there is no x in Z such that f(x)=x^{2}= -3

\therefore  f is not onto i.e. not surjective.

Hence, f is neither injective nor surjective.

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