Given that  $f\left(x\right)=\frac{e^{\left|x\right|}-e^{-x}}{e^x+e^{-x}}$

If $x>0,\left|x\right|=x$

$f\left(x\right)=\frac{e^x-e^{-x}}{e^x+e^{-x}}=Tanhx$

If $x<0,\left|x\right|=-x$

$f\left(x\right)=\frac{e^{-x}-e^{-x}}{e^x+e^{-x}}=0$

$\therefore \text{'}f\text{'}$ is not one –one

Range = $\left(-1,1\right)$

Co-domain=R

Range$\ne$ Co-domain

$\therefore \text{'}f\text{'}$ is not onto

$\therefore \text{'}f\text{'}$ is neither one-one nor onto