### Instruction: Find the domain of the function
\[g(x) = \frac{x^3 + 11x - 2}{|x - 3| + |x + 1|}.\].### Response: To find the domain of a function, I need to identify all the values of x that make the function well-defined and meaningful.
The only potential problem I can see here is if the denominator becomes zero, because then I would have a fraction with zero in the bottom, which is undefined.
So I need to find out when |x - 3| + |x + 1| = 0.
The absolute value function is always nonnegative, so the only way for the sum of two absolute values to be zero is if both of them are zero.
That means I need to solve the system of equations
\[x - 3 = 0\]
\[x + 1 = 0.\]
This system has no solution, because x cannot be both 3 and -1 at the same time.
Therefore, there is no value of x that makes the denominator zero, and the function is defined for all real numbers.
The domain of g(x) is the set of all real numbers, or (-infinity, infinity) in interval notation.