0

I don't necessarily need an answer to this particular case, but in general I have no idea how to solve this kind of problem or even how to Google for such a method. If it helps for this particular problem, the context in which this expression came up only positive integer values of x $\mathit{really}$ makes sense, but if possible I would delight in finding a function that works for any real number.

(Apologies for the lack of tags, I have no idea what field this would fall under)

amWhy
  • 209,954
Rilazy
  • 13
  • Start by citing the source of your title question, and do so as an edit to your post. – amWhy Jan 03 '22 at 00:50
  • @amWhy The source of my title question is a tangent I followed off of my own recreational attempt to compute $\sqrt{2}$ – Rilazy Jan 03 '22 at 00:54
  • You can literally pick any function on $[0,1)$ and define $f$ inductively using your equation on $[0,\infty).$ It is harder if you need continuity, but not much harder. But if you need $f$ defined on all of $\mathbb R$ then you get some problems. – Thomas Andrews Jan 03 '22 at 01:02
  • 1
    "I don't necessarily need an answer to this particular case, but in general I have no idea how to solve this kind of problem or even how to Google for such a method." Sorry, but this site is not intended to be a tutoring site. It is a question and answer site, yet you are asking us to teach you how to solve "this kind of problem" which you told me is actually tangent", and also how to Google. – amWhy Jan 03 '22 at 01:02
  • 1
    There are a lot of sorts problems “like this.” From simple recursive functions, to general functional equations. You don’t state the domain of $f,$ so we don’t know if the $x$ are natural numbers, integers, real or complex. We don’t know if you have any additional conditions on $f,$ such as “is continuous.” – Thomas Andrews Jan 03 '22 at 01:06
  • To answer the question as asked: this is a functional equation. You can find useful information here: https://artofproblemsolving.com/wiki/index.php/Functional_equation – Alon Amit Jan 03 '22 at 01:19

1 Answers1

3

This is a functional equation. Such equations are ubiquitous in recreational math and math competitions, but also in “serious” math research – but then they are more likely to carry further restrictions such as being continuous or differentiable.

AoPS has a nice introduction to the topic. If you really need to solve this particular equation, you’ll want to be very specific about the domain and the codomain (natural numbers or real numbers, with or without 0), because they makes a whole world of difference.

Alon Amit
  • 15,591
  • 1
    This is a two links-only post, and hence not an answer. – amWhy Jan 03 '22 at 00:58
  • 10
    @amWhy the OP asked what domain such a question belongs to, and I answered that. I’m not sure why you say this is not an answer. – Alon Amit Jan 03 '22 at 01:00
  • 10
    @amwhy this condescending tone is entirely uncalled for, especially since I also used the term domain in that sense in my answer. – Alon Amit Jan 03 '22 at 01:06
  • I referred only to your use in a reply to me. And there are two links which do all the work for you. Your own words do not answer the question. Hence, it is a two-links-only answer. – amWhy Jan 03 '22 at 01:11
  • 2
    Well, for a post to count as an answer, there typically is mathematical reasoning in the post that solves the math problem stated by the OP. A post where the main contribution is a link or two to a reference, is generally considered on here to be more appropriate as a comment.¹ – Mike Jan 03 '22 at 01:13
  • 1
    @amwhy my own words including just what the OP was looking for: a term (“functional equation”) they can Google or otherwise research. If I had not included any links would you consider the answer legitimate? This is genuinely confusing. – Alon Amit Jan 03 '22 at 01:15
  • 3
    @Mike I understand, I’ve been using math.SE too infrequently to be aware of the etiquette. Happy to remove my answer and add it as a comment on the question instead. – Alon Amit Jan 03 '22 at 01:17
  • 1
    Well, what I think would be needed for this to be an answer, would be answering w a mathematical proof that establishes whether an $f$ that satisfies the OP's conditions exist, where the domain is as in the OP's choosing e.g., rationals, real line, positive reals, etc – Mike Jan 03 '22 at 01:19
  • 3
    @Mike apparently I cannot delete an accepted answer. – Alon Amit Jan 03 '22 at 01:20
  • 1
    Alon yes, this is more appropriate as a comment. In all fairness to you, the OP was somewhat vague in what he was looking for. But yes in general, "answers" are to solve the mathematical question at hand, unless the subject tag is "Reference Request" – Mike Jan 03 '22 at 01:25
  • 2
    Alon I did not mean to insult you. Perhaps I should have elaborated as Mike did. I was sincerely trying to inform and hoped your would elaborate on your answer. And I agree with Mike that the OP was very vague. And in such cases, it is often best to encourage the asker to clarify, and refrain from answering until the asker does so. I know you only sought to help. – amWhy Jan 03 '22 at 01:26
  • 2
    I don't think its a big deal @Alon Amit. For next time... – Mike Jan 03 '22 at 01:28