Smarter people than I have written entire books about the function of art. My attempt, here -- which will probably come across as overly simplified -- will be to hit the highlights. Disclaimers abound.
First, proceed with this caution: No piece of art can be "assigned" a function (or functions), either in essay form or in casual conversation, if it isn't first considered within the proper context. Trying to classify function depends on context.
Ideally, one can look at a piece and know (approximately) where it came from and when. The best-case scenario includes identifying the artist, as well, because s/he is half of the contextual equation (i.e.: What was the artist thinking at the time s/he created this?). You, the viewer, are the other half (i.e.: What does this piece of art mean to you, living right now?). These are all factors that should be considered before trying to assign functions. Besides, taking anything out of context can lead to misunderstanding, which is never a happy place to visit.
That said, the functions of art normally fall within three categories. These are personal, social or physical functions. These categories can, and (often) do, overlap in any given piece of art.
The physical functions of art are most easily dealt with. Works of art that are created to perform some service have physical functions.
If you see a Fijian war club you may assume that, however wonderful the craftsmanship may be, it was created to perform the physical function of smashing skulls.
A Japanese raku bowl is art that performs a physical function in the tea ceremony. Conversely, a fur-covered teacup from the Dada movement has no physical function.
Architecture, any of the crafts and industrial design are all types of art that have physical functions.
Art has social functions when it addresses aspects of (collective) life, as opposed to one person's point of view or experience.
For example, public art in 1930s Germany had an overwhelming symbolic theme. Did this art exert influence on the German population? Decidedly so. As did political and patriotic posters in Allied countries during the same time.
Political art (skewed to whatever message) always carries a social function. The fur-covered Dada teacup, useless for holding tea, carried a social function in that it protested World War I (and nearly everything else in life).
Art that depicts social conditions performs social functions. The Realists figured this out early in the 19th century. Dorothea Lange (and, indeed, many other photographers) often photographed people in conditions we'd rather not think about.
Additionally, satire performs social functions. Francisco Goya and William Hogarth both went this route, with varying degrees of success at enacting social change.
Sometimes having specific pieces of art in a community can perform the social function of elevating that community's status. A Calder stabile, for example, can be a community treasure and point of pride.
The personal functions of art are the most difficult to explain in any great detail. There are many of them, and they vary from person to person.
An artist may create out of a need for self-expression, or gratification. S/he might have wanted to communicate a thought or point to the viewer. Perhaps the artist was trying to provide an aesthetic experience, both for self and viewers. A piece might have been meant to "merely" entertain others. Sometimes a piece isn't meant to have any meaning at all.
(This is vague, I know. The above is a great example of how knowing the artist can help one "cut to the chase" and assign functions.)
On a slightly more lofty plane, art may serve the personal functions of control. Art has been used to attempt to exert magical control over time, or the seasons or even the acquisition of food. Art is used to bring order to a messy and disorderly world. Conversely, art can be used to create chaos when an artist feels life is too staid and ordinary. Art can also be therapeutic - for both the artist and the viewer.
Yet another personal function of art is that of religious service (lots of examples for this, aren't there?). Finally, sometimes art is used to assist us in maintaining ourselves as a species. Biological functions would obviously include fertility symbols (in any culture), but I would also invite scrutiny of the ways we adorn ourselves in order to be attractive enough to, well, mate.
You, the viewer, are half of the equation in assigning function to art - as mentioned earlier. These personal functions apply to you, as well as the artist. It all adds up to literally billions of variables when trying to figure out the personal functions of art. My best advice is to stick with the most obvious and provide only those details you know as factual.
In sum, try to remember four points when required to describe "the functions of art": (1) context and (2) personal, (3) social and (4) physical functions. Good luck, and may your own words flow freely!