The given problem asks for the equation of the new function after applying two transformations to the cube root parent function $F(x) = \sqrt[3]{x}$:

1. Translate the function 1 unit to the right.
2. Translate the function 1 unit down.

Step-by-Step Solution:

1. Translation 1 Unit to the Right:

   • When you translate a function to the right by $h$ units, you replace $x$ with $x - h$. So for $h = 1$, the equation becomes: [ F(x) = \sqrt[3]{x} \rightarrow \sqrt[3]{x - 1} ]

2. Translation 1 Unit Down:

   • When you translate a function down by $k$ units, you subtract $k$ from the entire function. For $k = 1$, the equation becomes: [ \sqrt[3]{x - 1} \rightarrow \sqrt[3]{x - 1} - 1 ]

Thus, the new function $G(x)$ is: $G(x) = \sqrt[3]{x - 1} - 1$

Answer:

The correct option is B: $G(x) = \sqrt[3]{x - 1} - 1$.

Would you like further details or have any other questions?

Here are 5 related questions to expand your understanding:

1. What happens when a function is translated to the left instead of the right?
2. How would you graph the function $G(x) = \sqrt[3]{x - 1} - 1$?
3. What if the translation was 2 units right and 3 units down? What would the new function be?
4. How does the graph of the cube root function compare to that of the square root function?
5. What is the effect of reflecting the cube root function over the x-axis?

Tip: When translating functions, horizontal translations affect the input (inside the function), while vertical translations affect the output (outside the function).