verify that the functions of f and g are inverse of each other by showing that f(g(x))=x and g(f(x))=x; f(x)=In(x-1),g(x)=1+e^x

Answer 1

what's the problem? Just plug and chug:

f(g) = ln(g-1) = ln(1+e^x-1) = ln(e^x) = x
g(f) = 1+e^f = 1 + e^(ln (x-1)) = 1 + x-1 = x

Just keep in mind the definition of a logarithm:

e^lnx = x
ln e^x = x