Final answer:
The device used to establish a broadband connection from a home network to an ISP using a telephone line is a modem.
Step-by-step explanation:
The device used to make a broadband connection from your home network to the Internet Service Provider (ISP) using the telephone line is a modem. Modems have evolved from their origins in the 1960s and significantly increased in speed during the 1990s with the growth of the internet. Initially connecting via standard phone lines, the need for more data capacity led to the use of broadband connections. These modern modems enable high-speed internet access by modulating and demodulating the data sent over the telephone lines, incorporating technological advancements to provide reliable and fast internet connectivity.