Anti-theft Controller Problems of Hyundai Tiburon

I went to the car lot and picked up my Tiburon. I drove it home. . . And to work. After I had driven the car for 3 days, I walked toward the car and the wireless remote on the keys wouldnt beep so when I put the key in the door to unlock it the alarm sounded and the ignition would not start. I immediately contacted the dealer in which they said I probably needed to buy a new battery. I hadn't even put 100 miles on the car. I had someone check the battery and that wasn't the problem. My car sat at my workplace for 8 days before the dealership finally sent a towing truck to have it serviced. I come to find out it is the "transponder" to the alarm that is faulty and therefore the car won't start until it is replaced. It is still waiting to be serviced. My question is why do I have to pay to have that fixed when I have an extended warranty and I've only had the car a few days? they probably already knew it was going out and sold me the car anyway. I've only put down a down payment for the car and have yet to make my first payment. What should I do? are they right to charge me for this? is this a factory recall or something that goes out over time? or in this case right when I got the car? any help would be greatly appreciated. Thanks.

See all problems of the 1998 Hyundai Tiburon 🔎.