Final answer:
The worst-case time complexity for adding an item to the beginning of an array-based list is O(n), as it requires shifting each element in the array by one position to make space.
Step-by-step explanation:
When considering an array-based list, adding an item to the beginning of the list usually involves shifting all the existing elements one position forward to make space for the new element. The worst-case time complexity for this operation is O(n), where n is the number of elements in the array. This is because each element in the array must be moved once, hence the operation requires time proportional to the size of the array.
For example, if you have an array of 5 elements [1, 2, 3, 4, 5] and you want to add a new element at the beginning, you would need to move all existing elements: the 1 would move to the second position, the 2 to the third position, and so on until you make room at the beginning. Once the space is created at the index 0, you can insert the new element. This process involves touching each of the existing elements exactly once.
However, if the array is already full and cannot accommodate more elements, you may first need to create a new array with additional capacity, then copy the elements over after inserting the new element at the front. This might further increase the complexity of that specific operation, but the fundamental complexity for just adding to the beginning in the typical case remains O(n).