I'll elaborate more. In this situation, we are calling x + y the "input" of the function. This means that, no matter what, x + y will be in the correct domain--i.e. it will be a valid input for the function.
It's necessary to look at what an inverse function means. But to do that, we have to be sure you know what a function is. A function is something that you give an input to, and it gives you an output. Sine is a function. We give it an angle, and it gives us a number between -1 and 1. The inverse of sine is arcsine, but what does that mean? In a sense, it means that arcsine "undoes" whatever sine does. Let's look at some concrete example that don't involve these functions first.
Consider the function "Reverse phone book". Yes, this is a function. Why? Because we give it an input, and it assigns the input exactly one output. I give it the number 555-5555, and it outputs the name "John Smith". What would the inverse function look like? It would just be "phone book". I give it a name (John Smith), and it outputs a number (555-5555). (there is a complication with using this analogy--just assume for now that there are no repeated names in the book, and it will not be important for now).
So let's apply the same logic to sine. Sine takes in an angle, and it gives us a number between -1 and 1. So applying our logic, we see that the inverse function (arcsine) should take in a number between -1 and 1, and it should output an angle. The domain is just another name for the "list of valid inputs", so by extension right here, what's the domain?
Note: We limit the output to numbers between -pi/2 and pi/2, because otherwise, each input would have an infinite number of different outputs, and that's a no-no for functions.
