Have a telephone line that goes into the answering machine/ Also I have a fax mode on a printer. Currently I disconnecting the telephone jack from the answering machine and plug it into the printer with the mode on fax machine and send and receive faxes.Then unplugging and returning it to the answering machine.

Isn't there any due splitter I can put on whereby I do not have to disconnect
The answering machine to send a fax.

Having a split jack, and you can buy them, won't do you any good. You have to have a dedicated fax line; or basically a phone line and a separate one with a different number for your fax machine.

Yes you are right. Had the answering machine set up to receive a fax, after it hears the fax signal otherwise it answered the phone call.. But it confused my wife.

And when the fax is coming in, the telephone has a busy signal. So you are missing calls, and possibly when on the phone, missing faxes because there is a busy signal on the fax. So I hope this isn't a business fax.