f: Z → Z given by f(x) = x^3 - Vidyadip

Question

$f: Z → Z$ given by $f(x) = x^3$

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Answer

$f: Z → Z$ is given by,$f(x) = x^3$
It is seen that for $\text{x},\text{y}\in\text{Z},$
$f(x) = f(y) \Rightarrow x^3 = y^3 \Rightarrow x = y.$
$\therefore f$ is injective. Now, $2\in\text{N}.$
But, there does not exist any element $x$ in domain $Z$ such that $f(x) = x^3 = 2.$
$\therefore f $ is not surjective.
Hence, function f injective but not surjective.

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