ISI2014-DCG-24

Question

Let $f(x) = \dfrac{2x}{x-1}, \: x \neq 1$.  State which of the following statements is true.

• A. For all real $y$, there exists $x$ such that $f(x)=y$
• B. For all real $y \neq 1$, there exists $x$ such that $f(x)=y$
• C. For all real $y \neq 2$, there exists $x$ such that $f(x)=y$
• D. None of the above is true

Answer 1

$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 $f(x)=y$