The given function is $f(x) = \frac{1}{\sqrt[3]{x^2 - 2x - 3}}$.

To find the domain of the function, we need to ensure that the denominator is not equal to zero, as division by zero is undefined. So, we must solve the equation:

$x^2 - 2x - 3 = 0$

This is a quadratic equation, which can be factored as:

$(x - 3)(x + 1) = 0$

Thus, the solutions are $x = 3$ and $x = -1$.

These are the values of $x$ that make the denominator zero. Therefore, the function is undefined at $x = 3$ and $x = -1$.

Since we are dealing with a cube root, there are no further restrictions regarding the sign of the expression inside the root because cube roots can take negative values. Thus, the domain excludes only $x = 3$ and $x = -1$.

The correct domain is:

$\mathbb{R} - \{3, -1\}$

Thus, the correct answer is d) $R - \{ -1, 3 \}$.

Would you like more details or have any questions? Here are some related questions you may want to explore:

1. How do you solve quadratic equations by factoring?
2. What is the difference between square roots and cube roots in terms of domains?
3. How would the domain change if the root was even, like a square root?
4. How do you determine if a function is continuous at a certain point?
5. What are other methods to find the domain of more complex functions?

Tip: Always check for values that make the denominator zero or lead to undefined operations (like negative numbers under even roots).