Question

$f:R\to (-\infty ,1)$ is given by $f\left(x\right)=1-2^{-x},$ find the inverse of $f\left(x\right)$

Answer 1

Let $y=1-2^{-x}$

$y-1=-2^{-x}$

$1-y=2^{-x}$

$lo{g}_2(1-y)=-xlo{g}_{2}2$

$lo{g}_2(1-y)=-x$

$x={\log }_2\left(\frac{1}{1-y}\right)$

$\therefore f^{-1}\left(x\right)=lo{g}_2\left(\frac{1}{1-x}\right)$.