Hmm, the expression you wrote is equivalent to (8/8) + ex = 1 + ex. I think you meant:
ex is defined for all values of x. The only restriction on the domain, then, would be when ex = -8 since that would make the denominator equal zero and division by 0 is undefined.
ex = -8
x = ln(-8)
You can't take the log of a negative number, so there is no value of x such that ex = -8. Hence the expression 8/(8+ex) is defined for all x. In interval notation: (-∞,∞)