"To investigate this question" means to thoroughly explore or examine the question in order to gather information, evidence, or answers. Full definition
The authors were limited, however, in how deeply they could investigate questions of racial or gender gaps in publishing. (sciencemag.org)
Researchers investigated this question in a study of people in dating relationships recruited online (average age of participants = 24 years old; 90 % heterosexual). (scienceofrelationships.com)
This study investigates this question by looking for «Social Dollars» — economic returns to firms from customers that can be attributed to the act of joining firm - specific online communities. (sciencedaily.com)