+595 Let
$$f(x) = \frac{1}{1+\frac{2}{1+\frac 3x}}.$$
There are three real numbers $x$ that are not in the domain of $f(x)$. What is the sum of those three numbers?
+2653 For f(x) to be defined, the inner term 1+x32 must be defined. This inner term is undefined when its denominator, 1+x3=0, so when x=−31. However, even at x=−31, the outer term 1+21 is still defined, so f(x) is actually defined at x=−31.
Therefore, the only restriction on the domain of f(x) comes from the denominator of the entire expression, 1+1+x32. This denominator is undefined when its denominator, 1+x3, is 0. So the function is undefined when x=−31, as discussed above, and also when x=−1.
Finally, note that as x approaches positive or negative infinity, the entire expression approaches 11, which is a defined value. Therefore, the only restrictions on the domain are x=−31 and x=−1. The sum of these two numbers is −31−1=−34.
