How can functionalism and conflict perspectives work together?
Functionalism is an approach to sociology based on the idea that social institutions are structures which groups of people build in order to achieve certain functions, such as maintaining order or making profit. It focuses on how institutions and societies hold themselves together. One of its founders was Emile Durkheim.
Conflict theory is an approach to sociology based on the idea that social change is driven by conflicts between different groups of people, particularly groups based on race, sex, and economic class. Originally based on the works of Karl Marx, conflict theory postulates that competition between groups over scarce resources is at the heart of all social structures.
These two perspectives sound very different, even diametrically opposed; but they can really be thought of as two extremes of a continuum. At one end, functionalism says that society is based upon cooperation to achieve shared goals; at the other end, conflict theory says that society is based upon conflict over scarce resources. Reality is of course some combination of the two; obviously conflicts do arise between different groups of people (on various scales, ranging from individual families to whole nations), but at the same time there is cohesion within most groups most of the time, and not all conflicts are resolved violently. People do often cooperate with one another for common goals.
One place where I think the two perspectives are both necessary is in understanding war. Plainly, war involves a great deal of violent conflict over scarce resources, so we need conflict theory; but that's not the whole story. In order to fight a war as opposed to just a few random guys fighting over a piece of land, you need an enormous amount of organization and cooperation toward particular functions. A modern war effort mobilizes the labor of millions of people in concert, integrating skills from hundreds of different disciplines---a modern war needs physicists and economists at least as much as it needs pilots and snipers. To understand how such a complex system can hold itself together at a time of crisis, we need functionalism.