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HP Forum Archive 16

 What would YOU call this?Message #1 Posted by Chuck on 22 Dec 2006, 12:31 p.m. Last week a few friends and I discussed this "function": f(x)=ln(1-x)+ln(x-2) It was seen in a classroom, to which the instructor had written, "the domain is empty, therefore this is NOT a function". On further thinking, though, it does not violate any of the parts of the definition of a function. Sooooo, what would you call it? I have my own idea(s). p.s. Stay out of the complex's. :)

 Re: What would YOU call this?Message #2 Posted by Valentin Albillo on 22 Dec 2006, 12:40 p.m.,in response to message #1 by Chuck Hi, Chuck: I guess you want this: ``` f(x) = ln(1-x)+ln(x-2) = ln((1-x)*(x-2)) = ln(-2+3*x-x2) where the argument of ln() is positive for x in (1,2), so f(x) is defined and real for x in that range, i.e between 1 and 2, both excluded, with extrema at x=1.5 ``` Best regards from V.

 Re: What would YOU call this?Message #3 Posted by Chuck on 22 Dec 2006, 12:48 p.m.,in response to message #2 by Valentin Albillo Ahh, but your first step is not allowed; you have drastically changed the function. The original function f(x) = ln(1-x)+ln(x-2) cannot be evluated at x = 1.5 without delving into the complexes. It's like saying Log[x^2] = 2Log[x]. The graphs of these are clearly not identical (for x>0 they are, but for x<0 they are not). Soooo, is the original a function or not?

 Re: What would YOU call this?Message #4 Posted by John Gustaf Stebbins on 22 Dec 2006, 3:19 p.m.,in response to message #1 by Chuck Looking at a couple definitions of "function" I guess you could say that it is a function on the empty set, assuming you consider the empty set to be a valid domain. Doesn't seem proper, but I don't see where the laws of mathematics would fall apart.

 Re: What would YOU call this?Message #5 Posted by Crawl on 23 Dec 2006, 6:47 a.m.,in response to message #1 by Chuck I'd call it a function, because "stay out of the complex" is an artifical human requirement, while analytic continuity is mathematically natural. I'd also say x / x = 1 at x = 0 (not that it's undefined), and 1 - 1 + 1 - 1 + 1 -... = 1/2, though. If you want to have a function that has no domain, why not define a function that really has no domain? Or at least something weirder than the logarithm, which is a perfectly normal function, except that it's multi-valued. How about f(x) = 1^x + 1^(2*x) + 1^(3*x) + 1^(4*x) + 1^(5*x) + ... which diverges to infinity for all x. Or even use something non-mathematical. ``` x, if x loves 0 f(x) = x^2, if x doesn't love 0 ``` f(x) could take values, but since it's hard to establish if one number loves another, it might be undefined for all x.

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