Question

If the function $f:R\to A,$ given by $f\left(x\right)=\frac{e^x-e^{-\left|x\right|}}{e^x+e^{\left|x\right|}}$ is surjection,then $A\in [m,k.)$,Find 2(m+k) ?

Answer 1

Given $f:R\to A,$ is defined by
$f\left(x\right)=\frac{e^x-e^{-\left|x\right|}}{e^x+e^{\left|x\right|}}$
Since, f is surjection,
So, $R\left(f\right)=\text{Codomain A}$
When $x>0$
$f\left(x\right)=\frac{e^x-e^{-x}}{e^x+e^x}$
$\Rightarrow f\left(x\right)=\frac{e^{2x}-1}{2e^{2x}}$
Let $y=\frac{e^{2x}-1}{2e^{2x}}$ ....(1)
$\Rightarrow e^{2x}=\frac{1}{1-2y}$
$\Rightarrow x=\frac{1}{2}\log \left(\frac{1}{1-2y}\right)$
$\Rightarrow \frac{1}{1-2y}>0$
$\Rightarrow y<\frac{1}{2}$
By (1), $y>0$ for $x>0$
Hence, range of f is $[0,\frac{1}{2})$
So, $k=0,m=\frac{1}{2}$
$\Rightarrow 2(k+m)=1$