Question

The function $f:R\to R$ defined by $f(x)=\frac{e^{|x|}-e^{-x}}{e^x+e^{-x}}$ is

• A. one-one and onto
• B. one-one but not onto
• C. not one-one but onto
• D. neither one-one nor onto

Answer 1

Answer: (D)

Given, $f(x)=\frac{e^{|x|}-e^{-x}}{e^x+e^{-x}}$
For $x<0,$ $f(x)=\frac{e^{-x}-e^{-x}}{e^x+e^{-x}}=0$
Here, we see that for all negative values of x, we get always zero, it means it is not for one-one. For
$x\ge 0,$
$e^{|x|}>e^{-x}$
$\therefore$ For $x>0,f(x)>0$
and for $x<0,f(x)=0$
Hence, no negative real value of
$f(x)$ exist, Hence, it is not onto.