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

Question

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

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Answer

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

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