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To find the domain of the function \( g(x) = \frac{3}{\sqrt[3]{x^2 - 3x + 2}} \), we need to consider the properties of the cube root and the denominator:

1. **Cube Root**: The cube root function, \( \sqrt[3]{x} \), is defined for all real numbers, meaning that the expression inside the cube root doesn't impose any restrictions on the domain by itself.

2. **Denominator**: However, we must ensure that the denominator is not equal to zero because division by zero is undefined. Therefore, we need to solve:
   \[
   x^2 - 3x + 2 = 0
   \]
   Factorizing the quadratic equation:
   \[
   (x - 1)(x - 2) = 0
   \]
   So, \( x = 1 \) and \( x = 2 \) are the values that make the denominator zero, which means \( x = 1 \) and \( x = 2 \) must be excluded from the domain.

### Conclusion:
The domain of the function is all real numbers except \( x = 1 \) and \( x = 2 \), meaning the domain is:
\[
(-\infty, 1) \cup (1, 2) \cup (2, +\infty)
\]
This corresponds to option **D**.

Would you like further details on this? Here are some additional questions to expand the understanding:

1. How would the domain change if it were a square root instead of a cube root?
2. Can you describe why \( x = 1 \) and \( x = 2 \) are excluded from the domain?
3. What happens to the function's graph around \( x = 1 \) and \( x = 2 \)?
4. How do cube roots differ from even roots in terms of domain restrictions?
5. Could there be any vertical asymptotes for this function?

**Tip:** Always factor the denominator first to identify critical values that must be excluded from the domain.

El dominio máximo de la función g de criterio g(x) = \frac{3}{\sqrt[3]{x^2 - 3x + 2}} corresponde a

Learn how to determine the domain of the function g(x) = 3 / (Cube Root of (x^2 - 3x + 2)). This problem involves understanding the properties of cube roots and the factorization of quadratic equations. Suitable for students in Grades 10-12.

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