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The function given is:

\[
g(x) = \frac{\sqrt[3]{x}}{x^2 + 9}
\]

To determine the domain of this function, we need to consider the following:

1. **Numerator Analysis:** The numerator is \( \sqrt[3]{x} \). The cube root function is defined for all real numbers, so there are no restrictions from the numerator.

2. **Denominator Analysis:** The denominator is \( x^2 + 9 \). Since this is a quadratic function \( x^2 \) plus a positive constant (9), it is always positive for all real numbers. Therefore, \( x^2 + 9 \) never equals zero, and there is no restriction on \( x \) from the denominator.

Since both the numerator and the denominator are defined for all real numbers, the function \( g(x) \) is defined for all \( x \in \mathbb{R} \).

**Conclusion:** The domain of \( g(x) \) is all real numbers. In interval notation, this is:

\[
\text{Domain of } g(x): \ (-\infty, \infty)
\]

Would you like further details on this problem, or do you have any questions?

Here are five related questions to consider:
1. What is the range of the function \( g(x) \)?
2. How would the domain change if the numerator were \( \sqrt{x} \) instead of \( \sqrt[3]{x} \)?
3. What happens to the domain if the denominator includes a variable term in the numerator (e.g., \( x^2 - 9 \))?
4. How do you find the domain of a function that includes a square root in the denominator?
5. What are the asymptotes, if any, of the function \( g(x) \)?

**Tip:** When analyzing the domain of a function, always consider where the function could become undefined, such as division by zero or even roots of negative numbers.

Find the domain of the function g(x) = (³√x) / (x² + 9).

Explore how to find the domain of the rational function g(x) = (³√x) / (x² + 9). This problem covers key mathematical concepts, including the domain of functions, real numbers, and the behavior of cube roots and quadratic expressions. Suitable for high school students in Grades 10-12.

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