Question

Find the range of the following functions
$\frac{x}{x-1}$

Answer 1

$f\left(x\right)=\frac{x}{x-1}$

Let, $y=\frac{x}{x-1}$

$⟹x=y(x-1)$

$⟹x=xy-y$

$⟹y=xy-x$

$⟹y=x(y-1)$

$⟹x=\frac{y}{y-1}$

$⟹f^{-1}\left(y\right)=\frac{y}{y-1}$

$\therefore f^{-1}\left(x\right)=\frac{x}{x-1}$

$Range=R-\left\{1\right\}$