The "symbol shaped like U" is the setunion: is the set of all members of both sets A and B. It is, simply, a way ofcombiningvarious parts of an answer. I am surprised that you had not learned that long before taking pre-calculus. And, by the way, the term is "piece-wise", not "peace-wise".

For -3< x<= 2, f(x)= -4 (you don't mention a function "f" but without one what you have written makes no sense at all) so one possible value for f(x) is -4.

For 2< x< 4, f(x)= 2x- 4 so that f(x) can take on values between 2(2)- 4= 0 and 2(4)- 4= 4 so f(x) can take on all values of x from 0 to 4.

For 4<= x< 7, f(x)= -(1/3)x+ 2 (you wrote "-1/3x+2" which most people would interpret as -1/(3x)+ 4, but I doubt that you mean that) so f(x) can take on all values between -(1/3)(4)+ 2= -4/2+ 6/3= 2/3 and -(1/3)(7)+ 2= -7/3+ 6/3= -1/3.

The domain of the entire function is the "union" of those three sets of number.