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Question

Check the injectivity and surjectivity of the following functions:$$f: Z \rightarrow Z$$ given by $$f(x)=x^2$$


Solution

$$f(x)=x^2$$
It is seen that $$f(-1)=f(1)=1$$, but $$-1 \neq 1$$
$$\therefore f$$ is not injective.
Now, $$-2 \in Z$$. But, there does not exist any

element $$x\in Z$$ such that $$f(x)=x^2=-2$$

$$\therefore f$$ is not surjective.
Hence, function $$f$$ is neither injective nor surjective.

Mathematics
RS Agarwal
Standard XII

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