Question

For $x\in R$, the range of $f\left(x\right)=\frac{2^x-1}{2^x+1}$ is

• A. $R$
• B. $R-[-1,1]$
• C. $(-1,1)$
• D. $(-1,1]$

Answer 1

The correct option is C $(-1,1)$

Let $y=\frac{2^x-1}{2^x+1}$
$\Rightarrow 2^{x}y+y=2^x-1\Rightarrow y+1=2^x(1-y)$
$\Rightarrow 2^x=\frac{y+1}{1-y}$

As $2^x>0\forall x\in R$,
So,
$\frac{y+1}{1-y}>0$
$\Rightarrow \frac{y+1}{y-1}<0$
$\Rightarrow y\in (-1,1)$

Hence, $R\left(f\right)=(-1,1)$