To determine which of the given options functions as both an input and output device, we must understand the roles of each type of device.

• Projector: A projector only receives data and displays it by projecting images onto a screen. Thus, it solely functions as an output device.
• Printer: A printer receives data from a computer and prints this information onto paper. As it only outputs data, it is considered purely an output device.
• Touchscreen Monitor: A touchscreen serves a dual purpose; it displays output like a regular monitor and also allows users to interact directly with what is displayed on the screen, acting as an input device. For example, touching the screen to open an application or typing on an on-screen keyboard.
• Keyboard: A keyboard is strictly an input device as it sends input data (keystrokes) to a computer without receiving any output.

From the examination of the functions above, the Touchscreen monitor is the only device that performs both input and output operations. Therefore, the correct answer is a touchscreen monitor.