Question

Find the range of the following functions
$f\left(x\right)=\frac{2+x}{2-x}$

Answer 1

$f\left(x\right)=\frac{2+x}{2-x}$

Let, $y=\frac{2+x}{2-x}$

$⟹2+x=y(2-x)$

$⟹2+x=2y-xy$

$⟹x+xy=2y-2$

$⟹x(1+y)=(2y-2)$

$⟹x=\frac{2y-2}{y+1}$

$⟹f^{-1}\left(y\right)=\frac{2y-2}{y+1}$

$\therefore f^{-1}\left(x\right)=\frac{2x-2}{x+1}$

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