# How do you find the inverse of f(x)=2^(x+1) -2?

Dec 16, 2015

$\textcolor{w h i t e}{\times} {f}^{-} 1 \left(x\right) = {\log}_{2} \left(\frac{x + 2}{2}\right)$

#### Explanation:

$\textcolor{w h i t e}{\times} f \left(x\right) = {2}^{x + 1} - 2$

$\implies y = {2}^{x + 1} - 2$
$\implies y \textcolor{red}{+ 2} = {2}^{x + 1} - 2 \textcolor{red}{+ 2}$

$\implies {2}^{x + 1} = y + 2$
$\implies x + 1 = {\log}_{2} \left(y + 2\right)$
$\implies x + 1 \textcolor{red}{- 1} = {\log}_{2} \left(y + 2\right) \textcolor{red}{- 1}$

$\implies x = {\log}_{2} \left(y + 2\right) - 1$
$\implies x = {\log}_{2} \left(\frac{y + 2}{2}\right)$

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