I know that functions can be their own inverse such as $f(x)=x$ however I thought there were only two $f(x)=x$ and $f(x)=-x$. Is there more?
What are the functions that are their own inverse?  (thus functions where $ f(f(x)) = x $ for a large domain)  I always thought there were only 4:  $f(x) = x , f(x) = -x , f(x) = \frac {1}{x} $ and $ f(x) = \frac {-1}{x} $   Later I heard about a fifth one     $$f(x) = \ln\left(\frac {e^x+1}{e^x-1}\right) $$ (for $x &gt; 0$)  This made me wonder are there more? What are conditions that apply to all these functions to get more, etc.
How do I list the members of a group? How can I see all the members of a group in Linux? Did spacetime start with the Big bang? Separability over intersection of intermediate fields Let $E\mathop{/}F$ be a finite-degree normal extension of fields. Let $K$ and $L$ be intermediate fields (that is, $F\subseteq K\subseteq E$ and $F\subseteq L\subseteq E$) such that $E$ is separable over both $K$ and $L$. Show that $E$ is separable over $K\cap L$.   So I was thinking that if $f=\min_{K\cap L}(\alpha)$. Then $\min_{K}(\alpha)$ divides $f$. But if $f$ has repeated roots then I don't see what goes wrong.   Thanks! Auto-fit TextView for Android New UI encourages askers to confirm or dispute duplicate votes Execution order of event receivers and workflow on a list Why is my StackOverflow with an indent and doesn't have reputation Case expressions in Access Fourier transform of the indicator of the unit ball Why does the series $\sum_{n=1}^\infty\frac1n$ not converge? Are there any banned subjects on Area 51? I know of the following rules for creating a proposal on Area 51:  The scope must be reasonably well-defined, not too unclear (e.g. &quot;cool stuff&quot;), too opinion-based (e.g. &quot;car reviews&quot;), too broad (e.g. &quot;life on Earth&quot;), or too narrow (e.g. &quot;The 1995 academic misconduct scandal at East Oak Mews High School&quot;). One  come to Area 51 with an already existing community rather than hope to build one on Area 51. Proposals that are subsets of existing sites are not acceptable (e.g.  — deleted). These get closed as &quot;This subject is already well-covered by a live Stack Exchange site. We generally do not split off subjects simply to give them their own space.&quot; Proposals for non-English sites .  Other than the criteria above, are there any per se restrictions on site content for an Area 51 proposal? If so, what topics are currently banned? I would expect a proposal for a &quot;Let's organize a lawsuit against Stack Exchange&quot; or &quot;[Specific SE staff member] sucks, amirite?&quot; would get closed for obvious reasons, but I'm interested if there are any non-obvious topics that are not currently allowed in proposals regardless of how well-defined, well-scoped, original, and well-supported by a community they are. Having a list of banned topics would be helpful to potential proposers as well as the communities that back them. For example, I wouldn't want to go to all the effort of organizing a community, bringing them to Area 51, etc., only to have my proposal closed with a curt &quot;Sorry, due to the political climate, we are not currently accepting proposals for sites about real-life weapons.&quot; custom closure. To be clear, I am asking solely about topic restrictions (e.g. &quot;no sites about fishing, hunting, or weapons&quot;). I'm not asking about clarity requirements, the requirement to have a pre-existing community, not duplicating an existing site, or the requirement to not use sock puppets to artificially inflate proposal stats. Those requirements are already well-defined elsewhere or else are common sense. Also to be clear, I don't have a specific proposal idea in mind right now or the community to make it happen. I'm asking in general. The question could be rephrased as &quot;What are those topics where I shouldn't even bother to try to define, gather a community, etc. because any attempt to propose a site will be summarily closed due to covering a banned topic?&quot; This is not a duplicate of  because that question is about the appropriateness of expanding a proposal to cover things that are acceptable proposal topics but arguably don't make sense in combination with the original proposal scope. My question is about topics that are completely banned from proposals regardless of whether they are included in the original proposal description or tacked on by the community later. Why are potential differences equal across two capacitors in parallel, but charge on each capacitor is not? Two or more capacitors are said to be connected in parallel if each one of them is connected across the same two points. In a parallel combination of capacitors potential difference across each capacitor is same but each capacitor will store different charge. Why is this true? Why is the potential difference across each capacitor in parallel the same and why will each capacitor store different charge?  Also, in series, why are Potential difference across each capacitor different, while charge is the same? Downloading from Play Store gives error 491 When I try to download anything from the Play Store it shows error 491 and can't finish the download. I have a Samsung Galaxy S 7582. How do I fix this? Why is this sequence of equicontinuous functions uniformly bounded? Let $\left\{f_{n}\right\}$ be a sequence of equicontinuous functions where $f_n: [0,1] \rightarrow \mathbf{R}$.  If $\{f_n(0)\}$ is bounded, why is $\left\{f_{n}\right\}$ uniformly bounded? How to create a GUID / UUID I'm trying to create globally-unique identifiers in JavaScript.  I'm not sure what routines are available on all browsers, how &quot;random&quot; and seeded the built-in random number generator is, etc. The GUID / UUID should be at least 32 characters and should stay in the ASCII range to avoid trouble when passing them around. Change multiple filenames by replacing a character I guess it is too difficult a question to ask about the cardinality of $S_{\mathbb{N}}$ so I would like to ask whether it is countable or not.  I tried to prove it is uncountable somewhat mimicking the Cantor's diagonal argument but failed. If $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x) dx = 0$ iff $f(x) = 0$  "$\Rightarrow$" Assume by contradiction that $f(x) \neq 0$ for some $x_0 \in [a, b]$. Without loss of generality, assume that $f(x_0) &gt; 0$. Since $f$ is continuous, $\exists \delta &gt; 0$ such that $f(x) &gt; 0,  \forall x \in (x_0 - \delta, x_0 + \delta)$. Since $f$ is continuous on $[a, b]$, then it is Riemann integrable. Therefore $\int_{a}^{b} f(x) dx = \sup L(f, p)$ for some partition $p$. Now what I want to do is to show that the lower sum, $L(f, p) &gt; 0$, in that case its supremum is $&gt;0$, and that contradicts the condition that $\int_{a}^{b} f(x) dx = 0$. However, I don't know how to articulate it.  "$\Leftarrow$" If $f(x) = 0, \forall x \in [a, b]$, then $\int_{a}^{b} f(x) dx = \int_{a}^{b} 0 dx = 0$ I solved the Lagrangian of a simple pendulum (with help from online examples as this concept is new to me) and ended with the following: $$\ddot{\theta}+\omega^2\theta=0$$ But in the example I was following, the next line written was this: $$\theta(t) = A\cdot\sin(\omega t + \Phi)$$ And I don't know what steps they took to get there? How to make git ignore changes in case? I'm not too sure what is going on here, but sometimes a particular file in my repository will change the case of its name. e.g.,: before: File.h after: file.h I don't really care why this is happening, but this causes git to think it is a new file, and then I have to go and change the file name back. Can you just make git ignore case changes? [edit] I suspect it is Visual Studio doing something weird with that particular file, because it seems to happen most often when I open and save it after changes. I don't have any way to fix bugs in VS however, but git should be a bit more capable I hope.