Question

The inverse of the function $y = \frac{2^x}{1+2^x}$ us :

• A. $x = \log_2 \frac{y}{1-y}$
• B. $x = \log_2 \frac{y}{1-2^y}$
• C. $x = \log_2 (1 - \frac{1}{y})$
• D. $x = \log_2( \frac{1}{1-y})$

Answer 1

Answer: (A)

Given,
$y=\frac{2^{x}}{1+2^{x}}$
$ \Rightarrow \,y+2^{x} y=2^{x} $
$\Rightarrow \, y=2^{x}(1-y)$
$ \Rightarrow \, 2^{x}=\frac{y}{1-y}$
Taking log both side at base 2, we get
$\log _{2}\left(2^{x}\right)=\log _{2} \frac{y}{1-y}$
$\Rightarrow \, x \, =\log _{2} \frac{y}{1-y} $
$\therefore $ Inverse $ =\log _{2} \frac{y}{1-y}$