what you did is correct. it can be explained as follows:
1) the value of z completely depends on the value of x+y.
2)the lower the value of x+y the lower the value of z.
3) the least value of x+y available in D is 0.
so e^0-2 is the answer.
Here the situation was fairly simple so we could use the above reasoning. In general such problems are solved using multivariate calculus.