Let, $y=\frac{x}{1+x^2}$

$\Rightarrow y+y{x}^2-x=0$

$\Rightarrow x^{2}y-x+y=0$

$\Rightarrow x=\frac{+1\pm \sqrt{1-4y^2}}{2y}$

$\Rightarrow 1-4y^2 \ge 0$

$\Rightarrow 4y^2 \le 1$

$\Rightarrow$ So, $y\in \left[-\frac{1}{2},\frac{1}{2}\right]$

Here, Co-Domain$=$Range

Hence, the function is surjective.

Now, for injective function

$\Rightarrow \frac{x_1}{1+x_1^2}=\frac{x_2}{1+x_2^2}$

$\Rightarrow$$x_1+x_2{x}_2^2-x_2-x_2{x}_1^2=0$

$\Rightarrow$$\left(x_1-x_2\right)+x_1{x}_2\left(x_2-x_1\right)=0$

$\Rightarrow$$\left(x_2-x_1\right) \left(x_1{x}_2-1\right)=0$

$\Rightarrow$$x_2=x_1$ or $x_1=\frac{1}{x_2}$

Hence, the function is not injective.