The short answer to your question is yes.
Most internet providers do not change the IP address assigned to you, it only guarantee not to change if you pay for a public static ip. The ip address provided by your internet provider is your public ip address, which is use to access your website. Each device on a network has a ip address. In the post I will only be talking about the router having the pubic ip address and the server having a non-public ip address. You will setup port forwarding to for clients request to the non-public ip address, which can be static. http://help.filemaker.com/app/answers/detail/a_id/9770/~/port-forwarding-information. You can have a static non-public ip address on your local server, which does not cost anything. I will suggest using a static non-public address on your server. You will go to your network adapter setting. Instead of obtain the ip address from the network you will select use ip address and then enter a non-public ip address. (Example 192.168.1.77 ) You computer will most likely have an non public ip address assigned to it, you do not want this address changing every time you turn off your computer because if it does, then you would have to change port forwarding on your router each time. You can get your current non-public ip address by turning on sharing in filemaker and your ip address that is used to access your file will be displayed. This is the address you want static on your local computer.
You can access a website such as www.ipchicken.com to get your current public ip address. This is the address you use to access your website from outside your local network. You can access your website by your domain name. This name will be setup to forward to your public ip address. If you do not have a static ip address then you would have to change this with your domain name provider and this can take a day or two to update, which is the main reason for a static ip address, so your site will not be available by domain name for two or more days.
You can download a manual from your router manufacture to find out how to setup port forwarding on your router.
You might also call a local computer business in your area to set this up for you, if you feel like this is over your abilities. There is tons of information on the internet.
I didn't have time to reread this post, ran out of time, so I apologize for any typo's.
ISPs DO change the public ip address regularly. In any case there's no guarantee that it won't be changed. No-ip.com offers a free dynamic dns service. There are other companies as well. It's a good idea to subscribe to one of these. No ip checks your public ip every 15 minutes and redirects all traffic.
My ISP doesn't change regularly. Mine has been the same for over a year. I guess it would have to do with your provider.