
Functions: Domain and Range (page 2 of 2) Sections: Functions versus relations, Domain and range Let's return to the subject of domains and ranges. When functions are first introduced, you will probably have some simplistic "functions" and relations to deal with, being just sets of points. These won't be terribly useful or interesting functions and relations, but your text wants you to get the idea of what the domain and range of a function are. For instance:
The above list of points, being a relationship between certain x's and certain y's, is a relation. The domain is all the xvalues, and the range is all the yvalues. To give the domain and the range, I just list the values without duplication: domain: {2, 3, 4, 6} range: {–3, –1, 3, 6} (It is customary to list these values in numerical order, but it is not required. Sets are called "unordered lists", so you can list the numbers in any order you feel like. Just don't duplicate: technically, repetitions are okay in sets, but most instructors would count off for this.) While the given set does represent a relation (because x's and y's are being related to each other), they gave me two points with the same xvalue: (2, –3) and (2, 3). Since x = 2 gives me two possible destinations, then this relation is not a function. Note that all I had to do to check whether the relation was a function was to look for duplicate xvalues. If you find a duplicate xvalue, then the different yvalues mean that you do not have a function.
This is another example of a "boring" function, just like the example on the previous page: every last xvalue goes to the exact same yvalue. But each xvalue is different, so, while boring, this relation is indeed a function. In point of fact, these points lie on the horizontal line y = 5. There is one other case for finding the domain and range of functions. They will give you a function and ask you to find the domain (and maybe the range, too). I have only ever seen (or can even think of) two things at this stage in your mathematical career that you'll have to check in order to determine the domain of the function they'll give you, and those two things are denominators and square roots.
As I can see from my picture, the graph "covers" all yvalues (that is, the graph will go as low as I like, and will also go as high as I like). Since the graph will eventually cover all possible values of y, then the range is "all real numbers".
The graph starts at y = 0 and goes down from there. While the graph goes down very slowly, I know that, eventually, I can go as low as I like (by picking an x that is sufficiently big). Also, from my experience with graphing, I know that the graph will never start coming back up. Then the range is "y < 0".
The graph goes only as high as y = 4, but it will go as low as I like. Then: The range is "all y < 4". << Previous Top  1  2  Return to Index


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