Answer
a) $x\gt\ln3$
b)$f^{-1}(x)=\ln(e^x+3)$
Domain: all real x
Work Step by Step
a) $e^x-3\gt0$
$e^x\gt3$
$x\gt\ln3$
b) $y=f(x)=\ln(e^x-3)$
$e^x-3=e^y$
$e^x=e^y+3$
$x=\ln(e^y+3)$
$y\rightleftharpoons x$:
$f^{-1}(x)=y=\ln(e^x+3)$
$\because e^x+3\gt0$ for all real x, the domain is all real x
