The modem usually connects to the Time Capsule which connects to the computer, etc.—but, following the TC's demise, we had to connect the modem directly to the iMac so we still had internet access. Just no wi-fi. That meant only one of us had computer access at a time.
I checked the prices of both a new and a refurbished time Capsule. Replacing it would be pricey.
My electrical engineer hubby did bit of Googling and found we could get a refurbished one for about $190. But a bit more Googling told him that a lot of first generation Time Capsules died when they were 18 months old and that Apple replaces hardware connected to a computer that has Apple Care. Fortunately, I'd gotten the 3-year Apple Care service contract with the iMac.
I called Apple, was connected with a service person within three minutes, and was assured that Apple Care would take care of a replacement Time Capsule. I'd have to send the dead one back, though, or I'd owe Apple $300.
They Fed-Ex'ed us the replacement, we received it on Tuesday, and soon had wi-fi. I had to call Apple tech support for help in restoring the Time Machine backup function to the Time Capsule, but I was connected promptly and was walked through the set-up by a very patient tech rep.
We sent the dead one back Wednesday, and they received it yesterday. Both wi-fi and backup are working fine.
When in doubt, Google.