To represent the total number of miles \( M \) the car will have on the odometer after \( w \) weeks, we start with the current mileage of 121,358 miles and add the average weekly mileage accumulation of 255 miles multiplied by the number of weeks \( w \).
The correct function rule is:
\[ M(w) = 121,358 + 255w \]
This equation states that the total mileage \( M \) after \( w \) weeks is equal to the initial mileage plus the mileage added over those weeks.
Thus, the correct response is:
M(w) = 121,358 + 255w