# How do we find the inverse of the function y=e^(2x)/(7+e^(2x))?

Sep 22, 2016

Inverse function of $f \left(x\right) = y = {e}^{2 x} / \left(7 + {e}^{2 x}\right)$ is $y = \frac{1}{2} \ln \left(\frac{7 x}{1 - x}\right)$

#### Explanation:

Let $f \left(x\right) = y = {e}^{2 x} / \left(7 + {e}^{2 x}\right)$

i.e. $\frac{7 + {e}^{2 x}}{e} ^ \left(2 x\right) = \frac{1}{y}$ or

$\frac{7}{e} ^ \left(2 x\right) + 1 = \frac{1}{y}$ or

$\frac{7}{e} ^ \left(2 x\right) = \frac{1}{y} - 1 = \frac{1 - y}{y}$ or

${e}^{2 x} = \frac{7 y}{1 - y}$ or

$\ln \left(\frac{7 y}{1 - y}\right) = 2 x$ and hence

$x = \frac{1}{2} \ln \left(\frac{7 y}{1 - y}\right)$

Hence inverse function of $f \left(x\right) = y = {e}^{2 x} / \left(7 + {e}^{2 x}\right)$ is $y = \frac{1}{2} \ln \left(\frac{7 x}{1 - x}\right)$