Let $f(x) = \dfrac{2x}{x-1}, \: x \neq 1$. State which of the following statements is true.
$f(x)=\frac{2x}{x-1}=y $
$\Rightarrow 2x=yx-1 $
$\Rightarrow x=\frac{1}{y-2}$
Thus $y\neq2$ for x to exist.
Hence Option(C) for all real $y\neq2$, there exists $x$ such that $f(x)=y$