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### Problem:  
We are tasked with analyzing the function \( G(x) = \sqrt[3]{x(1 + x^3)} \) to determine if it is continuous everywhere in its domain using Theorems 4, 5, 7, and 9. We also need to state the domain of the function.

### Solution:

1. **Domain Analysis:**
   The function \( G(x) = \sqrt[3]{x(1 + x^3)} \) is a cube root, and cube roots are defined for all real numbers. Therefore, there are no restrictions on \( x \), and the function is defined for all \( x \in \mathbb{R} \). Thus, the domain of \( G(x) \) is:
   \[
   \text{Domain of } G(x) = \mathbb{R}
   \]

2. **Continuity Analysis using Theorems:**

   Let's apply the following theorems:

   - **Theorem 4** (Polynomials are continuous):
     The expression \( x(1 + x^3) \) is a polynomial because it is the product of two polynomials. Since polynomials are continuous everywhere, this polynomial is continuous on \( \mathbb{R} \).

   - **Theorem 5** (Root functions are continuous):
     The cube root function \( \sqrt[3]{x} \) is continuous for all real \( x \). Since the expression inside the cube root, \( x(1 + x^3) \), is continuous (as explained above), the composition of the cube root with this expression is also continuous for all \( x \).

   - **Theorem 7** (Composition of continuous functions):
     The function \( G(x) = \sqrt[3]{x(1 + x^3)} \) is a composition of the cube root function and the polynomial function \( x(1 + x^3) \). Since both functions are continuous, their composition is also continuous for all \( x \in \mathbb{R} \).

   - **Theorem 9** (Continuity on the domain):
     Since the function is continuous for all real numbers based on the theorems above, it is continuous at every point in its domain, which is all real numbers.

### Conclusion:

- **Domain:** \( \mathbb{R} \) (all real numbers)
- **Continuity:** The function \( G(x) \) is continuous for all \( x \in \mathbb{R} \) based on Theorems 4, 5, 7, and 9.

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Would you like more details on any step? Here are some related questions to explore:

1. How does Theorem 5 ensure continuity for root functions?
2. What is the difference between continuity at a point and continuity over an interval?
3. How does composition of functions affect their continuity?
4. Can cube root functions ever be discontinuous?
5. How do you determine the domain for functions involving roots and polynomials?

**Tip:** When analyzing the domain of a function involving roots, focus on the type of root and the domain restrictions of the underlying expression inside the root.

Explain, using Theorems 4, 5, 7, and 9, why the function G(x) = ∛x(1 + x³) is continuous at every number in its domain. State the domain.

Discover why the function G(x) = ∛x(1 + x³) is continuous at every number in its domain using key theorems on polynomial and root function continuity. Ideal for students in advanced placement math or college courses.

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