After checking with the algebra, we check visually with mapping diagrams and graphs.
Which do you find more convincing visually?
$(g \circ f)(x) = g(f(x)) =g(Ae^x+k) = \ln ( \frac
{(Ae^x + k)-k } A) = \ln(e^x) = x $.
$(f \circ g)(x) = f(g(x)) =f( \ln ( \frac {x-k}A) ) = A*e^{\ln ( \frac {x-k}A)} +k= A* \frac {x-k}A +k= x$.