f*f = integral from -inf to inf of f(t)f(x-t) dt = integral from -inf to inf of f(x-t)f(t)dt
The Attempt at a Solution
My question concerns some issues with part a.
I break the problem up into two cases.
Case 1: When rectangle is sliding into the rectangle, since f(x-t) is moving.
I expect it to be from -2 ≤ x ≤ 0
In setting up the integral, I'm unsure how to analyze it based off my notes.
What I currently have:
Integral from -1 to 0 of 1*(1+t) dt where 1 represents f(x-t) and (1+t) represents f(t) part.
Unfortunately, this results in the wrong solution, as it then equals 1/2
Clearly, "x" needs to be part of the bounds based on the answer. My problem is, I don't understand how to get the bounds. I have drawn the structure with the rectangle being stationary and then the other rectangle sliding through it. I am thinking that I need to get the bounds from this drawing.
Any insight here is what I need help with and likely will guide me to understanding this problem.
Case 2: When rectangle is sliding out of the rectangle.
I expect it to be from 0 ≤ x ≤ 2
The answer is
f*f(x) = x+2 if -2≤x≤0, = 2-x if 0≤x≤2, = 0 otherwise