Suppose and .

Note that the domain of is , the range of is , the domain of is all real numbers, and the range of is also all real numbers.

The composite function . For this function the domain would be all real numbers, but if does not exist then you can plug it in, and for to exist. So for this composite function the domain is .

The composite function . For this function the domain would be . Since all of these numbers are in the range of , they can be plugged in to .

My question is, what is the domain of the composite function ? I believe that it would be since you can't plug a negative into and you can't plug a number between and into the resultant composite function. Is this correct? Is there a better way to think about this that does not involve "unions" and "intersections"?