Question

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

Answer 1

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

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

$⟹x=y(1-x)$

$⟹x=y-xy$

$⟹x+xy=y$

$⟹y=x(1+y)$

$⟹x=\frac{y}{1+y}$

$⟹f^{-1}\left(y\right)=\frac{y}{1+y}$

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

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