The image you uploaded contains a mathematical expression and four options for the domain of a function.

The function presented in the image is:

$y = \frac{x^4}{12} - \frac{x^3}{3} - 9x + 1$

The question asks to identify the set on which the function is defined, meaning the domain of the function.

To determine the domain, we first observe the function is a polynomial, and polynomials are defined for all real values of $x$. Therefore, the domain is $(-\infty, +\infty)$.

Let’s look at the options:

1. $(-\infty, 0) \cup (2, +\infty)$
2. $(-\infty, +\infty)$
3. $(0, 2)$
4. $(-\infty, -2) \cup (0, +\infty)$

Since the polynomial is defined for all $x$, the correct answer is:

Option 2: $(-\infty, +\infty)$.

Let me know if you need further explanations or details! Would you like more questions related to this topic?

Related questions:

1. What are the general properties of polynomials and their domains?
2. Can rational functions have different domains?
3. How can you find the domain of a function with square roots or denominators?
4. What are the conditions for a function to be undefined?
5. How does the graph of a polynomial function behave for large values of $x$?

Tip: When analyzing a function’s domain, always check for potential restrictions, such as division by zero or square roots of negative numbers, which are common causes for domain issues.