Domain and Range normally are easier to determine by assuming ALL REAL NUMBERS and only then looking around for things that really are not included.

f(x) = sin(x)

All Real Numbers.

Nothing to throw out.

Done.

g(x) = 1/sin(x)

All Real Numbers

Wait, sin(x) can be zero. We had better avoid where k is an integer.

Done.

Okay, now you do one.