wave equation - method of reflection

Hey, I'm a bit confused on how to implement the method of reflection (is that the correct term?) on the wave equation.
The exercise is in the attached file.

My attempt:
I've extended f(x) to the whole line using even continuation. I guess I'm supposed to do the same thing for g(x).. so I did.
I know how to write the first term of u(x,y), but I'm having trouble with the second term. How is this integral can be simplified?

Also, when I did the continuation for f(x) I've drawn arrows (red) for each segment. the arrows represent the so called "direction" of the function (from 0 to 1). Is that always the case?

Thank you in advance






*sorry for the spelling mistakes (Happy)