So, to find the domain we have to see if our function imposes any restrictions onto our variable or in other words if there are values our function would not be defined for.
Now, the function ln(t) is only defined for t>0.
Here we have t = x+4. so we need x+4>0.
Simply subtracting 4 on both sides gives us x>-4. that mesns the Domain is D ={ x in R: x> (-4) }, where R are the real numbers.
To find the inverse we look at thr equation y = ln(x+4) and isolate x.
y=ln(x+4)
e^y = e^(ln(x+4))
e^y = x+4 as e and ln are inverse functions of each other
e^y -4 = x
Swapping x and y (just a notation thing really) we get that the inverse of f ist given by
f^(-1) (x) = e^x -4. (The 4 is not in the exponent)