Question

Let $f(x)=e^{\left\{e^{|x|}sgnx\right\}}$ and $g(x)=e^{[e^{\left[{x|}_{sgnx}\right]}},x\in R$ where \{\} and [] denotes the fractional and integral part functions respectively. Also $h(x)=\log (f(x))+\log (g(x))$, then for real $x,h(x)$ is

• A. An odd function
• B. An even function
• C. Neither an odd nor an even function
• D. Both odd as well as even function.

Answer 1

Answer: (A)

$h(x)=\log (f(x)\cdot g(x))$
$=\log e^{\{y\}+[y]}=\{y\}+[y]$
$=e^{|x|}sgnx$
$\therefore h(x)=e^{|x|}sgnx=\{\begin{array}{ll}{e}^x,& x>0\\ 0,& x=0\\ -e^{-x},& x<0\end{array}$
$\Rightarrow h(x)=\{\begin{array}{ll}{e}^{-x},& x<0\\ 0,& x=0\\ -e^x,& x>0\end{array}$
$\Rightarrow h(x)+h(-x)=0$ for all $x$.