What would YOU call this? « Next Oldest | Next Newest »

 ▼ Chuck Senior Member Posts: 320 Threads: 59 Joined: Dec 2006 12-22-2006, 12:31 PM 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. :) ▼ Valentin Albillo Posting Freak Posts: 1,755 Threads: 112 Joined: Jan 2005 12-22-2006, 12:40 PM 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. ▼ Chuck Senior Member Posts: 320 Threads: 59 Joined: Dec 2006 12-22-2006, 12:48 PM 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? John Gustaf Stebbins Junior Member Posts: 6 Threads: 0 Joined: Jun 2006 12-22-2006, 03:19 PM 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. Crawl Senior Member Posts: 306 Threads: 3 Joined: Sep 2009 12-23-2006, 06:47 AM 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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