Answer
(a) $(e,\infty)$
(b) ) $f^{-1}(x)=e^{e^{e^x}}$
Work Step by Step
Given the function $f(x)=ln(ln(lnx))$
(a) The domain requirement is that $ln(lnx)\gt0$ which leads to $lnx\gt 1$
one more step gives $x\gt e$, so the domain of $f$ is $(e,\infty)$
(b) ) To find the inverse, first let $y=ln(ln(ln x))$,
so $ln(ln x)=e^y$, which leads to $ln x=e^{e^y}$,
and one more step gives $x=e^{e^{e^y}}$,
switch $x,y$, we have $f^{-1}(x)=e^{e^{e^x}}$