Final answer:
A qualitative study would be appropriate to investigate patients' perceptions about quality of life after a diagnosis of liver cancer.
Step-by-step explanation:
The appropriate type of study to investigate patients' perceptions about quality of life after a diagnosis of liver cancer is a qualitative study. This type of study focuses on understanding people's experiences, thoughts, and perceptions through open-ended questions, interviews, or observations. In the case of liver cancer patients, a qualitative study could involve conducting interviews with patients to gather their perspectives on the impact of the diagnosis on their quality of life.