How to invert $f(x)=\dfrac{2^x}{1+2^x}$

Question

I know that if $$f(x)=\frac{2^x}{1+2^x}$$

then

$$f^{-1}(x)=\log_2 \frac{x}{1-x} $$

How can I show this?

Answer 1

$x = \frac{2^{y}}{1 + 2^{y}} \iff x(1 + 2^{y}) = 2^{y} \iff x + x2^{y} = 2^{y} \iff x = 2^{y} - x2^{y} = 2^{y} (1- x) \iff \frac{x}{1-x} = 2^{y} \iff y = \log_{2}(\frac{x}{1-x})$