Question 217256
You've done everything correctly so far. Now we're looking for {{{g^-1(4)}}}. If we replace x by 4 in your equation for the inverse we get:
{{{4-3=y+e^y}}}
which simplifies to
{{{1=y+e^y}}}
which leaves us with figuring out y. Other than some logic, combined with trial and error, the only way to find y that I can suggest is to look at the graph of g(x). (I'd suggest looking at the graph of the inverse but Algebra.com's graphing software will not work on {{{x-3=y+e^y}}}.)
{{{graph(600, 600, -8, 8, -8, 8, 3+x+e^x)}}}
Since the function's x's are the inverse's y's and vice versa and since we are looking for {{{g^(-1)(4)}}}, we can try to find where g(x) = 4. And from the graph it appears that g(0) = 4. And we can verify this by substituting 0 in for x and finding g(0) (which does indeed turn out to be 4). Since g(0) = 4 then {{{g^(-1)(4) = 0}}}.