If f( x )= a ^ a ^ | x | sgn x ; g( x )= a ^ [ a ^ | x | sgn x ] for a > 1 and x ∈ R…

If \(f(\mathrm{x})=\mathrm{a}^{\left\{\mathrm{a}^{|\mathrm{x}|} \operatorname{sgn} \mathrm{x}\right\}} ; g(\mathrm{x})=\mathrm{a}^{\left[\mathrm{a}^{|\mathrm{x}|} \operatorname{sgn} \mathrm{x}\right]}\) for a > 1 and
x ∈ R, where { } & [ ] denote the fractional part and integral part functions respectively, then which of the following statements hold good for the function h (x),
where (ln a) h (x) = (ln f (x) + ln g (x)).