Question

The range of the function $f\left(x\right)=\frac{e^x-e^{|x|}}{e^x+e^{|x|}}$ is

• A. $(-\infty ,\infty )$
• B. $[0,1)$
• C. $(-1,0]$
• D. $(-1,1)$

Answer 1

The correct option is C $(-1,0]$

$f\left(x\right)=\left\{\begin{array}{cc}\frac{e^x-e^x}{e^x+e^x},& x⩾0\\ \frac{e^x-e^{-x}}{e^x+e^{-x}},& x<0\end{array}\right\}$

$\Rightarrow f\left(x\right)=\left\{\begin{array}{cc}0,& x⩾0\\ \frac{e^{2x}-1}{e^{2x}+1},& x<0\end{array}\right\}$

$y=\frac{e^{2x}-1}{e^{2x}+1}$ $\Rightarrow e^{2x}=\frac{y+1}{1-y}$

This if for $x<0$

i.e $2x<0$ $\Rightarrow$ $e^{2x}<1$

$\Rightarrow 0<\frac{y+1}{1-y}<1$

$\Rightarrow y\in (-1,0)\forall x<0$

$\Rightarrow y\in (-1,0]$

$\Rightarrow Asy=0\text{for}x⩾0$

$y\in (-1,0]$

$\therefore \text{option}c\text{is correct.}$