Question

Let $f(x)=e^{\left\{e^{|x|} {sgn}\, x\right\}}$ and $g(x)=e^{\left[e^{\left[\left.x\right|_{{sgn} \,x}\right]}\right.}, x \in R$ 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)

$h(x)=\log (f(x) \cdot g(x))$
$=\log e^{\{y\}+[y]}=\{y\}+[y]$
$=e^{|x|} {sgn}\, x$
$\therefore h(x) = e^{|x|} sgn\,x = \begin{cases} e^x, & x > 0 \\[2ex] 0, & x = 0 \\[2ex] -e^{-x}, & x < 0 \end{cases}$
$\Rightarrow h(x) = \begin{cases} e^{-x}, & x < 0 \\[2ex] 0, & x = 0 \\[2ex] -e^{x}, & x > 0 \end{cases}$
$\Rightarrow h(x) + h(-x) = 0$ for all $x$.