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What is meant by the domain and range of a relation?
A relationship is a way of associating members of one set to members of another set (the two sets could be the same). The first of these sets is the domain and the second is the range.
Is it the domain the same as the independent variable?
Yes it is also called the manipulated variable. Y is the range and dependent
How would you use domain and range to determine whether a relation is a function?
Each element in the domain must be mapped to one and only one element in the range. If that condition is satisfied then the mapping (or relationship) is a function. Different elements in the domain can be mapped to the same element in the range. Some elements in the range may not have any elements from the domain mapped to them. These do not matter for the mapping to be a function. They do matter in terms of the function having an inverse, but that is an entirely different matter. As an illustration, consider the mapping from the domain [-10, 10] to the range [-10, 100] with the mapping defined by y = x2.
How do you state domain and range?
Domain is the set of all possible numbers for a function on the X axis on a graph, and range is the set of all possible numbers for a function along the Y axis on a grpah. (The X axis is the one that runs horizontally, while the Y axis runs vertically). The domain and range define from and up to which numbers a function's point (coordinate) may be located on a graph. To state the domain of a function, you must find out what values "x" may and may not be in the function (equation), and the same goes for range. A good way to check if you've got your domain and range right is to try plugging in the numbers that you have found to be "restricted" and see if they really do produce an impossible or inaccurate result, or doesn't give you a result at all!
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?
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).
