# How do you find the inverse of y = e^x/(1 + 4 e^x)?

May 15, 2018

$x = \setminus \ln \left(\setminus \frac{y}{1 - 4 y}\right)$

#### Explanation:

This question would be a "solving for the inverse of a rational functions question" and you would follow the same standard
procedure as you would for solving those equations.

First multiply both sides by $1 + 4 {e}^{x}$ :
$y \left(1 + 4 {e}^{x}\right) = {e}^{x}$
$y + 4 {e}^{x} y - {e}^{x} = 0$
$4 {e}^{x} y - {e}^{x} = - y$ , factor ${e}^{x}$
${e}^{x} \left(4 y - 1\right) = - y$
${e}^{x} = \setminus \frac{- y}{4 y - 1} = \setminus \frac{y}{1 - 4 y}$
$x = \setminus \ln \left(\setminus \frac{y}{1 - 4 y}\right)$