Question

If $f\left(x\right)=a^{\left\{a^{|x|}sgnx\right\}}$; g(x) = $a^{\left[a^{|x|}sgnx\right]}$ for $a>0$, a$\ne$ 1 and x $\in$ R, where { } & [ ] denote the fractional part and integral part functions respectively, then which of the following statements holds good for the function $h\left(x\right)$, where $\left(\ln a\right)h\left(x\right)=\left(\ln f\right(x)+\ln g(x\left)\right)$

• A. $h$ is even and increasing
• B. $h$ is odd and decreasing
• C. $h$ is even and decreasing
• D. $h$ is odd and increasing

Answer 1

The correct option is D $h$ is odd and increasing

$hx=\frac{\ln \left(f\left(x\right)g\left(x\right)\right)}{\ln a}=\frac{\ln \left\{a^{\left|x\right|}sgnx\right\}+\left\{a^{\left|x\right|}sgnx\right\}}{\ln a}$
$=\left\{a^{\left|x\right|}sgnx\right\}+\left[a^{\left|x\right|}sgnx\right]=a^{\left|x\right|}sgn=\{\begin{array}{c}axx>0\\ 0x=0\\ -{ax}^{-x}x<0\end{array}$
$\Rightarrow h\left(x\right)$ is odd function