Final answer:
The sum of all real numbers not in the domain of f(g(x)) is 2/3, obtained by considering the points where each function's denominator is zero and where g(x) equals the excluded value from the domain of f(x).
Step-by-step explanation:
The question involves finding the domain of the composition of two functions, f(g(x)). In order to do so, we need to identify values for which both the original functions f(x) and g(x) are undefined. These values occur when the denominators of each function are zero because division by zero is undefined in mathematics.
Function f(x) has a denominator of x+1, which means it is undefined when x = -1. Function g(x) has a denominator of 3x^2 + 11x + 10, and this quadratic equation factors to (3x+5)(x+2), so g(x) is undefined when x = -5/3 or x = -2.
When we compose these functions to find f(g(x)), we need to ensure that g(x) is not equal to -1, since this would make the denominator of f(x) equal to zero. This occurs when the numerator of g(x), which is the same as that of f(x), is zero, i.e., when 3x^2 - 10x - 25 is zero. By solving this quadratic equation, we find the zeros of g(x)'s numerator are 5 and -5/3. However, since -5/3 is already excluded from the domain of g(x), we only consider 5. Adding the excluded values -1 (from f(x)), -2, -5/3 (from g(x)), and 5 (when g(x) = -1), we get the sum of all real numbers that are not in the domain of f(g(x)) as -1 - 2 - 5/3 + 5 = 2/3.