First of all if the deck is going to be extended it will need an extra support.  The proper way would have been  to put it in 4 ft into the ground but the customer did not want to disturb the interlocking driveway.

2nd of all,  floor joists where are OK can be extended  and rotten joists replaced. Then some of the top boards will need to be replaced but this would be much cheaper than replacing the whole deck.

In this case  2 pieces of  2x10's were used to support the structure underneath.  They were glued and screws together , then a 2x8 was put in the top of those which was also glues and screwed to the  replaced and extended floor joists.  Then  a 2x12, was mounted to conceal the joint of the 2x8 and 2x10 and was also glued and screwed to the stucture.