Show that the function


Show that the function $f: Z \rightarrow Z: f(x)=x^{3}$ is one-one and into.



To prove: function is one-one and into

Given: $f: Z \rightarrow Z: f(x)=x^{3}$

Solution: We have,


For, $f\left(x_{1}\right)=f\left(x_{2}\right)$

$\Rightarrow \mathrm{x}_{1}^{3}=\mathrm{x}_{2}^{3}$

$\Rightarrow \mathrm{x}_{1}=\mathrm{x}_{2}$

When, $f\left(x_{1}\right)=f\left(x_{2}\right)$ then $x_{1}=x_{2}$

$\therefore f(x)$ is one-one


Let $f(x)=y$ such that $y \in Z$

$\Rightarrow y=x^{3}$

$\Rightarrow x=\sqrt[3]{y}$

If $y=2$, as $y \in Z$

Then we will get an irrational value of $x$, but $x \in Z$

Hence $f(x)$ is into

Hence Proved

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