Determine the value of:
a) g(-4)
b) g(-2)
c) g(1)
d) g(4)
The absolute maximum of g(x) occurs when x=....?
The absolute maximum of g(x) is...?
Drawing $\displaystyle f(x)$ out would help. You will see that your graph basically consists of line segments.
Recall that an integral represents the net area under the curve (from the curve to the x-axis). So all you're doing is calculating the areas of the rectangles (taking into account of the signs).
For example, to find $\displaystyle g(1) = \int_{-3}^{1} f(t) \ dt$, we want to find the area under the curve from $\displaystyle x = -3$ to $\displaystyle x = 1$
Since it's a piecewise function, we're going to have to break the integral up into two: $\displaystyle g(1) = \int_{-3}^{1} f(t) \ dt = \int_{-3}^{0} f(t) \ dt + \int_0^1 f(t) \ dt$
If you look at your graph, the two integrals corresponds to the area under each rectangle that is created. I assume you know how to find the area of a rectangle so hopefully you'll be good from here.