It's easy to get lured into the sweet deals of "Free Domain Name" when purchasing web hosting. Many web-hosting companies advertise such a free domain name.
The same is true for transfers. You'll get something for "free" if you transfer your domain name to your hosting company.
First, the domain name is never really free. You may get the first year for free, but after that, you'll pay full price for renewal every year. And that price may be well higher than what you would pay at a domain registrar. See what various webhosts really charge you for your domain here.
But that's not even the worst part...
The worst is this: if your domain name is controlled by your webhost, they can make it very hard for you to leave!
Unfortunately, this has happened more than once: customers of some fine web hosting company suddenly realize said hosting company has been acquired by another. Soon they cut costs, staff leaves (or is fired) and service degrades rapidly. Comes time of renewal, customers try to leave for another hosting company only to find that the company makes the procedure so convoluted they pass the due date for renewal. Renewal price is unexpectedly high... but hosting company holds the domain hostage and locks down your site. Customer has no choice but to pay through the nose...
Don't let this happen to you!
Pass on the one year of "free domain" and keep your domain free instead!
In case you wonder: some good domain name registrars are Namecheap Domains, Uniregistry and Moniker to cite a few.