Question

Let $f$ be a real valued function defined by $f(x)=\frac{e^x-e^{-|x|}}{e^x+e^{|x|}}$, range of $f$ is $[a, b)$, then find the value of $(5 a+4 b)$

Answer 1

$\Theta f(x)=\frac{e^x-e^{-|x|}}{e^x+e^{|x|}}=\begin{cases}\frac{e^x-e^{-x}}{2 e^x}, & x \geq 0 \\ 0, & x<0\end{cases}.$
$\therefore \text { for } x \geq 0 $
$ y = f ( x )=\frac{ e ^{ x }- e ^{- x }}{2 e ^{ x }}$
$ y =\frac{1}{2}\left(1- e ^{-2 x }\right) $
$\Theta 0 \leq x <\infty$
$\Rightarrow 0< e ^{-2 x } \leq 1 $
$\Rightarrow -1 \leq- e ^{-2 x }<0 $
$\Rightarrow 0 \leq 1- e ^{2 x }<1 $
$\Rightarrow 0 \leq \frac{1- e ^{2 x }}{2}<\frac{1}{2}$
$\therefore R _{ f }=\left[0, \frac{1}{2}\right) $
$\therefore 5 a +4 b =2$