Final answer:
Structural functionalism is the theory that asserts racial and ethnic inequalities have persisted because they serve an important role in maintaining social stability, despite being inherently problematic.
Step-by-step explanation:
The theory that suggests racial and ethnic inequalities must have served an important function in society is structural functionalism. This perspective posits that such inequalities exist because they contribute to social stability by reaffirming the roles and expectations of the dominant and subordinate groups. Functionalism views these disparities as integral parts of a well-functioning society, even if they promote social injustice. While this viewpoint can be seen as problematic, it helps analyze how racism and discrimination are justified and perpetuated by the dominant group, such as historically through the institution of slavery.