Rosita is writing an explicit function for the geometric sequence:

To determine the correct domain for the function \( t(n) = 160 \left( \frac{1}{2} \right)^n \) such that it generates the given geometric sequence \( 80, 40, 20, 10, \ldots \), we first need to consider how the sequence is structured.

The sequence starts with:
- \( t(0) = 80 \)
- \( t(1) = 40 \)
- \( t(2) = 20 \)
- \( t(3) = 10 \)

From this, we observe that:
- The first term \( t(0) \) corresponds to 80.
- The second term \( t(1) \) corresponds to 40.
- The third term \( t(2) \) corresponds to 20.
- The fourth term \( t(3) \) corresponds to 10.

For the function \( t(n) \) to generate the sequence correctly:
- We want \( n \) to start at 0 (producing the first term as 80) and can increase to any positive integer.

Therefore, the correct domain for \( t \) is:

**(Choice A)** \( n \geq 0 \) where \( n \) is an integer.

This allows \( n = 0, 1, 2, 3, \ldots \) which correctly matches the terms of the sequence. Choices that start with \( n \geq 1 \) would omit the first term of the sequence.

So, the correct answer is:
**(Choice A)** \( n \geq 0 \) where \( n \) is an integer.