Show that the function


Show that the function $f: N \rightarrow N: f(x)=x^{2}$ is one-one and into.


To prove: function is one-one and into

Given: $f: N \rightarrow N: f(x)=x^{2}$

Solution: We have,


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

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

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

Here we can't consider $x_{1}=-x_{2}$ as $x \in N$, we can't have negative values

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


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

$\Rightarrow y=x^{2}$

$\Rightarrow x=\sqrt{y}$

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

Then we will get the irrational value of $x$, but $x \in N$

Hence $f(x)$ is not into

Hence Proved


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