The inverse of $f(x)=\sqrt[3]{1-x^3}$ is itself As I was studying this function, $f(x)=\sqrt[3]{1-x^3}$, I checked that the function is one-to-one, and so is invertible.  Then: $$y=\sqrt[3]{1-x^3}$$  $$y^3=1-x^3$$  $$y^3-1=-x^3$$  $$1-y^3=x^3$$  $$x=\sqrt[3]{1-y^3}$$ $$f^{-1}(y)=\sqrt[3]{1-y^3}$$  What kind of functions are inverse of itselfs? Thanks.
Examples of involutions on $\mathbb{R}$ Just recently I read up something about involutions (functions $f: A \rightarrow A$ such that $f(f(x))=x$, for all $x$ in the domain of $f$), and was wondering how many (if there is a small set of general functions) such involutions exist for $A = \mathbb{R}$, or maybe $A = \mathbb{R} - S$, where S is some set of points that would make the involution work if they were left out of $A$. In general I'm interested in real functions that are involutions, continuous or otherwise.  There were some examples I found , but any more would be certainly very interesting!
What is butterfly lighting, and when do I use it? I often hear the term 'Butterfly Lighting' used when discussing high-fashion and glamor photography. How do I set up Butterfly Lighting, and when do I use it? Where do I ask a question about copyright? Show/Hide CSS Ignored When Inside an SLDS Grid Column Counting area percentage of overlapping polygons I have a set of points (meteorological stations) with 5km buffers around them. Additionally, I have a set of polygons (municipalities) in which those stations are located.  I would like to have data on how much of the buffer is filled by each polygon but I was not able to find a function or a way to do it. So for example for point A I would have that in the buffer of 5km there is 80% of polygon B and 20% of polygon C as the output.  Edit:  The anwsers provided do anwser this question in a setting in which I know what polygons and buffers I need to have in the code or when I can do it manualy. Problem is that in my setting with 1 shapefile having 500 polygons and one vector layer having 13000 buffers I am unable to do this manually and I am asking if there is an easier way to do it. I tried implementing the approaches in the questions but I was unable to use them on my example in big scale. Moreover I had problems with coding that I also noted in comments down below.   Main difference to the other similar question provided is that I have 2 vector layers (Buffer with 500 buffers and Municipalities a shape file with 500 different polygons). Previous questions were analyzing this question in situations in which I know precisely the polygons and buffers analyzed, at least that is how I understood them, so after trying to use those approaches on my datasets I was not able to replicate them.  Another way of approaching this problem for me is to use join attributes by location -one-to-many and then getting averages of the duplicate results, only problem is that then I take average of polygons instead of their weights corresponding to % of intersection with the buffer. In my case I do not know what polygons of the shapefile are going to intersect what buffers, or even if the buffer is going to be intersected by 1,2 or more polygons After Doomsday was defeated, a spec-ops team sent to the Kryptionian ship in Metropolis found Luthor submerged in water, in front of a... thing, which "held" three metal cubes. The thing looked up, roars, and dissipates into thin air.    What the hell was that all about? I am really confused. What was that creature and what was Luthor doing with it? Homemade coconut milk separates into different layers after a day in the fridge Integral of Sinc Function Squared Over The Real Line Is it "me" or "I" and why? Can Objective C determine the caller object reference at runtime (reflection)? When to use "If I was" vs. "If I were"? If I was... If I were...   When is it correct to use "If I was" vs. "If I were" in standard English? Neopronouns or trolling? It’s the  question: Am I looking at someone’s legitimate pronouns or trolling?   (It’s unfortunate I have to ask, but this is just where we are right now.)  I’ve seen “my pronouns are [insert ridiculous fake stuff here]” trolling recently (in posts) when before I never did. Also stuff that looks to be trolling, but could also maybe not be. Trolls deserve red flags while everyone else specifying pronouns does not, right?  (To clarify, the problem is not just about not wanting to use a troll's words. The purpose of some of this trolling is to try and convince people that transgender people—or their neopronouns—are stupid or unreasonable. Ignoring the trolls allows us to avoid the first problem, but not the second since it allows them to still reach people with their message, which is not something that I want to see promoted.)  Some neopronouns are clearly legitimate. Some trolling is clearly trolling. But unfortunately there’s a big grey area where it’s not clear to me what is what.   Does anyone have a good method for dealing with this? There just doesn’t even seem to be a good way to ask about someone’s specified pronouns in such a way that doesn’t sound insulting if the user isn’t trolling.     Note: A while ago, I did some research on neo-/third person pronouns which can be found . There are several important things I found:   There are more neopronouns than I could ever hope to keep track of The most popular neopronouns are very uncommon. On the other end of the spectrum, it looks like some neopronouns are neologisms used by a single person  Some neopronouns come from unlikely places, such as SciFi How to successfully put a bibliography in a report on TexShop? I'm trying to put a bibliography in my LaTeX document, but I get various errors when I run it in BibTeX (saying no \citation found etc.), and no bibliography is printed in the PDF when I typeset it in LaTeX. My code reads as follows:  \documentclass[12pt]{report} ... \usepackage{biblatex} \addbibresource{SUSYBib.bib} ... \begin{document} ...\cite{1}... ...\cite{2}... \nocite{*} \printbibliography \end{document}   And my .bib file reads as  @book{1, title={The Schrödinger Equation}, author={F. A. Berezin, M. A. Shubin}, year={1983}, publisher={Moscow State University, Moscow}, pages={57-59} } @book{2, title={An Introduction to Measure Theory}, author={Terence Tao}, year={2011}, publisher={American Mathematical Society}, pages={91} } Pixel Perfect 2D tiles I'm trying to make an 64x64 tilemap using Super Tilemap Editor &amp; Unity but my tiles looks terrible. I did my research about pixel perfect 2D and all of the solutions were for pixel art, they didn't work for me.  I have the following tileset I made in Adobe Illustrator as an example    Tiles looks like this when placed in game (1x zoom)    I'm pretty sure they're not 64x64 in game, they also look pixelated probably because they're smaller than they actually are on camera.  So my question is, how can I calculate the correct camera size for 64x64 tiles and do I have to make any changes in order to make sprites look pixel perfect? I'm currently using 1080/64/2=8.4375 (vertical resolution / PPU / 2) as camera size. Is there a way to output for Print. I want to save in a higher DPI than the default 72dpi say 300dpi that's a good quality for printed pieces. I tried downloading an add-on but don't know if I am installing it correctly it isn't showing up but the bar in python is green. I'm using the 2.69 version of Blender. If no such feature exists what size have any of you found best for print quality. Thank you Between you and ("me" or "I")? Control Android with broken screen from PC Why do I see a double variable initialized to some value like 21.4 as 21.399999618530273? double r = 11.631; double theta = 21.4;   In the debugger, these are shown as 11.631000000000000 and 21.399999618530273.  How can I avoid this? What is the difference between "why are you worried" and "why you are worried"? Read only field not working in page layout I have custom object called candidate. I have created field name called first name which is read-only in the page layout. Once a record is created with first name field populated I am able to edit that field. Why I am able to edit (I mean getting pencil icon). Can anyone suggest me for this issue? Unicode Regex; Invalid XML characters The list of valid XML characters is well known, as defined by the spec it's:  #x9 | #xA | #xD | [#x20-#xD7FF] | [#xE000-#xFFFD] | [#x10000-#x10FFFF]   My question is whether or not it's possible to make a PCRE regular expression for this (or its inverse) without actually hard-coding the codepoints, by using Unicode general categories. An inverse might be something like [\p{Cc}\p{Cs}\p{Cn}], except that improperly covers linefeeds and tabs and misses some other invalid characters.