A function is surjective, when the range of the function is same as the co-domain.

Now, $y=\frac{x^2}{1-x^2}$

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

$\Rightarrow y=x^2\left(y+1\right)$

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

$\because x^2\ge 0 \Rightarrow \frac{y}{y+1}\ge 0$

$\Rightarrow y \in \left(-\infty , -1\right)\cup \left[0, \infty \right)$ or $R-\left[-1, 0\right).$

$\therefore$ Range of $f=A=R-\left[-1, 0\right).$