### 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 the values of x that make the function undefined or invalid.
A common source of undefinedness is division by zero, so I should check if the denominator of the fraction can ever be zero.
The denominator is the sum of two absolute value expressions, |x - 3| and |x + 1|.
The absolute value of a number is always nonnegative, so the only way the denominator can be zero is if both |x - 3| and |x + 1| are zero simultaneously.
That means I need to solve the system of equations
\[\begin{cases}
|x - 3| = 0\\
|x + 1| = 0
\end{cases}\]
The absolute value of a number is zero if and only if the number itself is zero, so I can drop the absolute value signs and get
\[\begin{cases}
x - 3 = 0\\
x + 1 = 0
\end{cases}\]
Solving for x in both equations, I get
\[\begin{cases}
x = 3\\
x = -1
\end{cases}\]
However, these two values are inconsistent with each other, so there is no solution to the system of equations.
This means that there is no value of x that makes the denominator zero, so division by zero is not an issue for the domain of g(x).
Another possible source of invalidity is taking the square root of a negative number, but I don't see any square roots in the function, so I don't need to worry about that either.
Therefore, the domain of g(x) is all real numbers, since there are no restrictions on the values of x that make g(x) well-defined.