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

  1. For all real $y$, there exists $x$ such that $f(x)=y$
  2. For all real $y \neq 1$, there exists $x$ such that $f(x)=y$
  3. For all real $y \neq 2$, there exists $x$ such that $f(x)=y$
  4. None of the above is true
in Calculus
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1 Answer

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$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 $f(x)=y$

 

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