The function provided is:

$f(x) = \frac{x^2}{(x^2 - 1)(x - 3)}$

To find the domain of this function, we need to determine when the denominator equals zero because division by zero is undefined.

Step 1: Analyze the denominator

The denominator is:

$(x^2 - 1)(x - 3)$

Factor the expression $x^2 - 1$:

$x^2 - 1 = (x - 1)(x + 1)$

So the denominator becomes: $(x - 1)(x + 1)(x - 3)$

Step 2: Find the values that make the denominator zero

The denominator will be zero when any factor is zero: $x - 1 = 0 \implies x = 1$ $x + 1 = 0 \implies x = -1$ $x - 3 = 0 \implies x = 3$

Thus, the function is undefined at $x = 1$, $x = -1$, and $x = 3$.

Step 3: Domain

The function is defined for all real values of $x$ except $x = -1$, $x = 1$, and $x = 3$.

Conclusion:

The correct option is $x \neq -1, 1, 3$.

Do you need further details or have any questions?

Here are 5 related questions to explore:

1. What would happen if the numerator of the function were zero at one of the values that makes the denominator zero?
2. How would you handle finding the domain of a function with multiple variables?
3. What is the range of the given function?
4. How would you determine the vertical asymptotes for this function?
5. Can you explain how the domain of the function affects its graph?

Tip: Always factor the denominator fully to ensure you capture all restrictions on the domain.