Show that the function


Show that the function $f: R \rightarrow R: f(x)=1+x^{2}$ is many-one into.



To prove: function is many-one into

Given: $f: R \rightarrow R: f(x)=1+x^{2}$

We have,


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

$\Rightarrow 1+x_{1}^{2}=1+x_{2}^{2}$

$\Rightarrow x_{1}^{2}=x_{2}^{2}$

$\Rightarrow x_{1}^{2}-x_{2}^{2}=0$


$\Rightarrow x_{1}=x_{2}$ or, $x_{1}=-x_{2}$

Clearly $x_{1}$ has more than one image

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


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

$\Rightarrow y=1+x^{2}$

$\Rightarrow x^{2}=y-1$

$\Rightarrow x=\sqrt{y-1}$

If $y=3$, as $y \in R$

Then $x$ will be undefined as we can't place the negative value under the square root

Hence $f(x)$ is into

Hence Proved



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