I dont know if this helps but I hope it does. When I look for domain in a function I follow this simple check list that if there is a:
1. Fraction: Denominator cannot equal to zero. Most schools will require you to write 'all real x' as well as the x≠number.
2. Root: Everything under the root must be greater than or equal to zero (x≧no.). However, always remember that if a root lies within the denominator of a fraction that it CANNOT BE EQUAL TO the number .i.e. x>no.
3. Log: For logarithimic functions, if the qs is logx or log(x + 1) or whatever, the part that proceeds the log must be greater than zero, not equal to it .i.e. x>0 or x>-1, generally x>no.
This is my list for domain so far.
For finding out range, I like to draw a quick little sketch of it, especially,for functions such as parabolas or hyperbolas to see whether there are turning points or areas of discontinuity. If the qs requires a sketch, I always do range last, especially if it is those complex ones. Though usually if the qs tells u to find range, then it shouldn't be so hard to work out. Btw, if a y is equal to a square root of anything then the range would be y≧0.
Yeah.