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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
recategorized | 20 views

$f(x)=\frac{2x}{x-1}=y$

$\Rightarrow 2x=yx-1$

$\Rightarrow x=\frac{1}{y-2}$

Thus $y\neq2$ for x to exist.

Hence Option(C) for all real $y\neq2$, there exists $x$ such that $f(x)=y$

by Active (1.7k points)