Given, function $f\left(x\right)=\frac{2x}{x-1}$ and domain of $f\left(x\right)$ is $\left(-\infty , 0\right]$

Now $f^{\text{'}}\left(x\right)=\frac{\left(x-1\right)2-2x\left(1-0\right)}{{\left(x-1\right)}^2}$

$f^{\prime }\left(x\right)=\frac{-2}{{\left(x-1\right)}^2}<0$

$\Rightarrow f$ is always decreasing in the given domain.



Clearly, given function is one-one or injective.
Since, codomain $\left(-\infty , \infty \right)\ne$ Range $\left(0, 2\right)$ , hence $f$ is into.