Hello Makii,
Thank you for using Apple Support Communities!
Here is the information on the coverage you have on your battery with the iPhone:
Your one-year warranty includes replacement coverage for a defective battery. You can extend your coverage to two years from the date of your iPhone purchase with the AppleCare Protection Plan for iPhone. During the plan’s coverage period, Apple will replace the battery if it drops below 50% of its original capacity. If it is out of warranty, Apple offers a battery replacement for $79, plus $6.95 shipping, subject to local tax. Apple disposes of your battery in an environmentally friendly manner.
From: Battery Replacement
http://www.apple.com/batteries/replacements.html
I would also like to suggest some troubleshooting. These steps from the article:
iPhone: Hardware troubleshooting
http://support.apple.com/kb/TS2802
- Click ( www.apple.com/batteries/iphone.html) for tips on how to prolong iPhone's battery life.
- Try turning iPhone off and then on again.
- Connect iPhone to iTunes and restore iPhone.
- If the screen shows a low-battery image, the battery is low on power and needs to charge for up to ten minutes before you can use it.
- When charging iPhone, make sure to leave it charging until it is fully charged. You'll know the battery is fully charged when the battery icon in the upper-right corner of the screen looks like this .
Note: When charging using a computer, don't connect iPhone to a keyboard. Also, the computer must be turned on and not in sleep or standby mode. If iPhone is connected to a computer that's not turned on or is in sleep or standby mode, the iPhone battery may drain.
With regard to the restore, first I would backup the phone to iTunes, and restore from that backup. If the issue persists I would then restore it and set it up as new to see how that works.
All the best,
Sterling