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To determine the new function after translating \( f(x) = \sqrt[3]{x} \):

1. **Translation 3 units in the negative \( y \)-direction**:
   - Moving downward by 3 units translates the function vertically. This affects the function by subtracting 3 from \( f(x) \).
   - So, \( f(x) = \sqrt[3]{x} - 3 \).

2. **Translation 8 units in the negative \( x \)-direction**:
   - Moving left by 8 units translates the function horizontally. This affects the \( x \)-term inside the function, changing \( x \) to \( x + 8 \).
   - So, \( f(x) = \sqrt[3]{x + 8} - 3 \).

Therefore, the resulting function is:
\[
f(x) = \sqrt[3]{x + 8} - 3
\]

### Answer:
The correct choice is:
- **C) \( f(x) = \sqrt[3]{x + 8} - 3 \)**

Would you like a more detailed explanation of each translation? 

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Here are 5 related questions to further explore this topic:

1. How does translating a function affect its graph?
2. What happens if we translate a function in both positive \( x \)- and \( y \)-directions?
3. How would a reflection over the \( x \)-axis change the function \( f(x) = \sqrt[3]{x} \)?
4. How do we determine the transformations for different types of functions, such as quadratic or exponential functions?
5. What are the differences between vertical and horizontal transformations?

**Tip:** When translating functions, remember that changes inside the function (like \( x + h \)) affect the \( x \)-direction, while changes outside the function (like \( f(x) + k \)) affect the \( y \)-direction.

The function f(x) = ∛x is translated 3 units in the negative y-direction and 8 units in the negative x-direction. Select the correct equation for the resulting function.

This problem involves translating the cube root function f(x) = ∛x by 3 units in the negative y-direction and 8 units in the negative x-direction. The solution explores function transformations, focusing on vertical and horizontal translations. Suitable for high school students studying function transformations.

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