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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?

 Apr 15, 2024
 #1
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For f(x) to be defined, the inner term 1+x3​2​ 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+x3​2​. 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​​.

 Apr 15, 2024

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