The function of an argumentative essay is to show that your assertion (opinion, theory, hypothesis) about some phenomenon or phenomena is correct or more truthful than others'. The art of argumentation is not an easy skill to acquire. Many people might think that if one simply has an opinion, one can argue it successfully, and these folks are always surprised when others don't agree with them because their logic seems so correct. Argumentative writing is the act of forming reasons, making inductions, drawing conclusions, and applying them to the case in discussion; the operation of inferring propositions, not known or admitted as true, from facts or principles known, admitted, or proved to be true. It clearly explains the process of your reasoning from the known or assumed to the unknown. Without doing this you do not have an argument, you have only an assertion, an essay that is just your unsubstantiated opinion.
Notice that you do not have to completely prove your point; you only have to convince reasonable readers that your argument or position has merit; i.e., that it is somehow more accurate and complete than competing arguments.

Argumentative essays are often organized in the following manner:
They begin with a statement of your assertion, its timeliness, significance, and relevance in relation to some phenomenon.
They review critically the literature about that phenomenon.
They illustrate how your assertion is "better" (simpler or more explanatory) than others, including improved (i.e., more reliable or valid) methods that you used to accumulate the data (case) to be explained.
Finally revise and edit, and be sure to apply the critical process to your argument to be certain you have not committed any errors in reasoning or integrated any fallacies for which you would criticize some other writer.
Additionally, you will want to find out how your readers will object to your argument. Will they say that you have used imprecise concepts? Have you erred in collecting data? Your argument is only as strong as the objections to it. If you cannot refute or discount an objection, then you need to rethink and revise your position.