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 $(5a+4b)$

Answer 1

Answer: 2

$\Theta f(x)=\frac{e^x-e^{-|x|}}{e^x+e^{|x|}}=\{\begin{array}{ll}\frac{e^x-e^{-x}}{2e^x},& x\ge 0\\ 0,& x<0\end{array}.$
$\therefore \text{ for }x\ge 0$
$y=f(x)=\frac{e^x-e^{-x}}{2e^x}$
$y=\frac{1}{2}\left(1-e^{-2x}\right)$
$\Theta 0\le x<\infty$
$\Rightarrow 0<e^{-2x}\le 1$
$\Rightarrow -1\le -e^{-2x}<0$
$\Rightarrow 0\le 1-e^{2x}<1$
$\Rightarrow 0\le \frac{1-e^{2x}}{2}<\frac{1}{2}$
$\therefore R_f=\left[0,\frac{1}{2}\right)$
$\therefore 5a+4b=2$