### Instruction: Find the domain of the function
\[g(x) = \frac{x^3 + 11x - 2}{|x - 3| + |x + 1|}.\].### Response: The domain of a function is the set of all input values (x) for which the function is defined and produces a real output value (g(x)).
One thing that can make a function undefined is dividing by zero, so I want to avoid that.
To find the values of x that make the denominator zero, I need to solve the equation
\[|x - 3| + |x + 1| = 0.\]
The absolute value function is always nonnegative, so the only way this equation can be true is if both terms are zero.
That means I need to solve the system of equations
\[\begin{cases}
x - 3 = 0\\
x + 1 = 0
\end{cases}\]
This system has no solutions, because x cannot be both 3 and -1 at the same time.
Therefore, the denominator is never zero, and there are no restrictions from that.
Another thing that can make a function undefined is taking the square root of a negative number, but I don't see any square roots here, so I don't need to worry about that either.
There are no other operations that can cause problems for the function, so I conclude that the domain is all real numbers.
I can write that using interval notation as (-infinity, infinity), or using set-builder notation as {x | x is a real number}.