In words, what does the function g(x) represent on the graph above?
The function g(x) represents the area under the graph of f between a and x.
Thus, g(a) = 0, since the area between a and a is zero. According to the second fundamental theorem of calculus,
f (t)dt = f (x)
This tells us two things. First, an area function of f is always an antiderivative of f.
Second, differentiation and integration are inverse operations, because integrating f and
then differentiating it yields f again.