If the function


If the function $\mathrm{f}: \mathrm{R}-\{1,-1\} \rightarrow \mathrm{A}$ defined by $\mathrm{f}(x)=\frac{x^{2}}{1-x^{2}}$, is

suriective, then $A$ is equal to:

  1. (1) $\mathrm{R}-\{-1\}$

  2. (2) $[0, \infty)$

  3. (3) $\mathrm{R}-[-1,0)$

  4. (4) $\mathrm{R}-(-1,0)$

Correct Option: , 3



$\Rightarrow f(-x)=\frac{x^{2}}{1-x^{2}}=f(x)$

$f^{\prime}(-x)=\frac{2 x}{\left(1-x^{2}\right)^{2}}$

$\therefore \mathrm{f}(\mathrm{x})$ increases in $\mathrm{x} \in(10, \infty)$

Also $\mathrm{f}(0)=0$ and

$\lim _{x \rightarrow \pm \infty} f(x)=-1$ and $\mathrm{f}(\mathrm{x})$ is even function

$\therefore \operatorname{Set} \mathrm{A}=\mathrm{R}-[-1,0)$

And the graph of function $f(x)$ is

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