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

Question

f: $Z \rightarrow Z$ given by $f(x) = x^3$

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Answer

$f: Z \rightarrow Z$ 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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