Solving for the range of a multivariable function.

[Ascendant78]

I am currently taking Calc III as an online course (yes, big mistake). I am at a section where we are evaluating the domains and ranges for functions with both x and y variables in it.

As far as finding domains, no problem. However, the textbook doesn't explain how to solve for ranges with x and y components in a function very well. I am wondering if there's a specific method (steps) for doing so, or if it varies depending on the structure of the function?

The example they used was they first solved for the domain, then manipulated that domain equation to get a part of it to look like the restricted portion of the original function to find the range. While it wasn't too difficult for that particular problem, it just seems like it would get really sloppy and challenging to do with more complex problems. I'm just wondering if there is a specific method or if it is sort of like a related rate problem where you really have to just analyze it and figure it out?

 

[lurflurf]

It does get hard for more complicated functions. Usually only simple functions are considered. Sometimes the function can be broken into pieces which makes things easier. Various identities and simplifications are helpful. It helps to find the extreme values. If the function is continuous we know we can include all values between known value. Can you post some examples?

 

[Ascendant78]

It does get hard for more complicated functions. Usually only simple functions are considered. Sometimes the function can be broken into pieces which makes things easier. Various identities and simplifications are helpful. It helps to find the extreme values. If the function is continuous we know we can include all values between known value. Can you post some examples?

Well, the examples so far I've figured out, like restrictions based on arcsin, square roots, denominators, etc. You pretty much answered my questions already anyway that it all seems to depend on the specific function and what rules you can apply to restrictions. I just wanted to make sure there wasn't an easier method or formula you could use to do them.

I'm assuming for more complicated functions, we would just use a graphing program anyway. I'm also assuming that in grad school, I wouldn't waste my time figuring out a functions graph myself and would just plug it into something like Mathematica, but I could be wrong of course. Well, thanks for the feedback, I appreciate it.