I had to google "full width half max". Apparently, it is the distance between and for points such that 1/2 the maximum value of the function.
You have correctly calculated that the maximum value of this function occurs at x= 0 and, so, is e. But the "half max" is not 1/e, it is, as you should guess, half of e: e/2.
You need to solve . Since f is symmetric about x= 0, the "full width at half maximum" is twice that x value.