When you compose two functions you must know the domain and range of the original functions to find the domain and range of their composition?

Answer 1

This is kind of a tricky question. First off, the range of a function is not what you're after. You want the codomain. The range of a function is the set of all of the values that are possible as a result of the function acting on every element in the domain. The codomain, in contrast, is more generally thought of as where the function was constrained to fall in the first place, prior to even knowing what the function was.

Think of a game of pool. When you take a shot, the range of where the cue ball will end up (assuming you don't scratch) is on the table. The codomain, however, is the entire three dimensional room. The range constraint of the codomain was due to the function which mapped the ball from its starting point to it's functionally allowed ending point. In this case, the function could be called "Legal Billiard Shot." However, the function could have been, "Throw Cue Ball At Friend's Head" which would have had the same exact codomain, the three dimensional area of the room, but a completely different range.

Now for the actual answer to your question. When composing two functions, say f: x --> y and g: y --> z, which yields g(f(x)) --> z, what you actually need to know is only the codomain of f(x) and only the domain of g(y), and they have to be the exact same set. You can't take a composite function if you can't be guaranteed that the range of the first function, which is a subset of it's codomain, is also a subset of the domain of the second function, ie: every value, y, has to be able to produce an actual, definable result when acted on by g(y).