To determine the domain of the function $f(x) = \sqrt[3]{13x^2 - 12x}$, we need to identify the set of all $x$ values for which this function is defined.

Key Points:

1. Cube Root Properties:

   • The cube root $\sqrt[3]{a}$ is defined for all real values of $a$. Unlike square roots, cube roots are valid for both positive and negative arguments.

2. Expression Inside the Cube Root:

   • The expression $13x^2 - 12x$ inside the cube root is a polynomial, which is defined for all real values of $x$.

Thus, the function $f(x)$ is defined for all real numbers.

Domain:

The domain of $f(x)$ is: $(-\infty, \infty)$

Let me know if you have any questions or would like additional explanations!

Follow-Up Questions:

1. What are the differences in domain rules for cube roots versus square roots?
2. How does the presence of a fraction affect domain analysis in other cases?
3. How would the domain change if the cube root were replaced by a square root?
4. Can you graph this function to see its behavior over different $x$-values?
5. How would you handle a domain restriction if there were additional conditions (e.g., division by zero)?

Tip:

Always start domain analysis by identifying restrictions like square roots (must be non-negative) or denominators (cannot be zero).