Cartesian Equations-domain and range

https://www.quora.com/profile/Bravewarrior/p-146695284
Here is the question and its solution. I get the mark scheme but was basically wondering if anyone knew a better way to work out the domain and range for these kind of questions as I kind of struggle with this bit. Thanks!

The worked solutions are pretty much the way to go. About the only thing Id suggest is sketching the basic functions (and simple transformations) for x(t) and y(t) so you can visualise the algebra/reasoning about the domain/range