What is the domain of the function
`f(x)=4-sqrt(3x^(3)-7)`?

Options

A. `xge1.33`
B. `xge1.53`
C. `xge2.33`
D. `xle-1.33` or `xge1.33`

Explanation

To find the domain of the function \( f(x) = 4 - \sqrt{3x^3 - 7} \), we need to ensure that the expression under the square root is non-negative, as the square root of a negative number is not defined in the real number system.

### Step-by-Step Solution:

1. **Identify the condition for the square root:**
We need the expression inside the square root to be greater than or equal to zero:
\[ 3x^3 - 7 \geq 0 \]

2. **Rearrange the inequality:**
To isolate the term involving \( x \), we can add 7 to both sides:
\[ 3x^3 \geq 7 \]

3. **Divide by 3:**
Since 3 is a positive number, we can divide both sides of the inequality by 3 without changing the direction of the inequality:
\[ x^3 \geq \frac{7}{3} \]

4. **Take the cube root:**
To solve for \( x \), we take the cube root of both sides:
\[ x \geq \sqrt[3]{\frac{7}{3}} \]

5. **Calculate the cube root:**
The value of \( \frac{7}{3} \) is approximately \( 2.3333 \). Thus, we can find:
\[ \sqrt[3]{\frac{7}{3}} \approx 1.442 \]

6. **State the domain:**
Therefore, the domain of the function \( f(x) \) is:
\[ x \geq \sqrt[3]{\frac{7}{3}} \quad \text{or} \quad [\sqrt[3]{\frac{7}{3}}, \infty) \]

### Final Answer:
The domain of the function \( f(x) = 4 - \sqrt{3x^3 - 7} \) is \( [\sqrt[3]{\frac{7}{3}}, \infty) \).