When a board, commission, committee, council, or other agency created by or pursuant to this act was preceded by an agency with the same or similar name and functions, members of the predecessor agency shall continue in office for the duration of the terms of office for which they were appointed and with the new members appointed shall constitute the new agency. Members shall be appointed under this act only as terms of the former members expire or vacancies occur. Members of the predecessor agency may be appointed to the new agency to succeed themselves subject to the limits for the total period of service set forth in this act.