The functions of each pair are inverse to each other. For each pair, check that both compositions give the identity function.  $F: \mathbb{R} \to \mathbb{R}$ and $F^{−1}:\mathbb{R} \to \mathbb{R}$ are defined by $F(x)=3x+2$ and $F^{−1}(y)=\dfrac{y−2}{3}$, for all $y \in \Bbb{R}.$  My attempt:     Inverse Function  For each particular but arbitrarily chosen $y \in \mathbb{R}$, according to the definition of $f^{-1}$, $f^{-1}(y) = \dfrac{y-2}{3}$ is a unique real number $x$ such that $f(x) = y$. \begin{align*} F(x) &amp; = y\\ 3x + 2 &amp; = y\\ x &amp; = \frac{y-2}{3} \end{align*} Therefore, $f^{-1}(y) = \frac{y-2}{3}$.  Compositions of Functions.   The functions $g \circ f$ and $f \circ g$ are defined as follows: $$(g \circ f)(x) = g(f(x)) = g(3x + 2) = 3x + 2$$  for all $x \in \mathbb{Z}$.
The functions of each pair are inverse to each other. For each pair, check that both compositions give the identity function.  $F: \mathbb{R} \to \mathbb{R}$ and $F^{−1}:\mathbb{R} \to \mathbb{R}$ are defined by $F(x)=3x+2$ and $F^{−1}(y)=\dfrac{y−2}{3}$. for all y ∈ R  My attempt:     Inverse Function  For each particular but arbitrarily chosen $y \in \mathbb{R}$, according to the definition of $f^{-1}$, $f^{-1}(y) = \dfrac{y-2}{3}$ is a unique real number $x$ such that $f(x) = y$. \begin{align*} F(x) &amp; = y\\ 3x + 2 &amp; = y\\ x &amp; = \frac{y-2}{3} \end{align*} Therefore, $f^{-1}(y) = \frac{y-2}{3}$.  Compositions of Functions  The functions $g \circ f$ and $f \circ g$ are defined as follows: $$(g \circ f)(x) = g(f(x)) = g(3x + 2) = 3x + 2$$  for all $x \in \mathbb{Z}$.
Computer doesn't boot up, shows PXE error and/or "operating system not found", "no bootable device", "insert boot media", or other similar error My computer no longer boots Windows or the installed operating system. Instead, I may get this error:  PXE-E61: Media test failure, check cable -or- Check cable connection!  PXE-M0F: Exiting PXE ROM.   and/or one of the following messages or similar:  Operating System not found  No boot device available- No bootable devices--strike F1 to retry boot, F2 for setup utility  No bootable device -- insert boot disk and press any key  No Boot Device Found. Press any key to reboot the machine  Default Boot Device Missing or Boot Failed  Reboot and Select proper Boot device or Insert Boot Media in selected Boot device   What does this mean and what can I do?     This question comes up often and the answers are usually the same. This post is meant to provide a definitive, canonical answer for this problem. Feel free to edit the answer to add additional details. Proving inequality $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{3\sqrt[3]{abc}}{a+b+c} \geq 4$ I started to study inequalities - I try to solve a lot of inequlites and read interesting .solutions . I have a good pdf, you can view from  . The inequality which I tried to solve and I didn't manage to find a solution can be found in that pdf but I will write here to be more explicitly.  Exercise 1.3.4(a) Let $a,b,c$ be positive real numbers. Prove that $$\displaystyle \frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{3\sqrt[3]{abc}}{a+b+c} \geq 4.$$ (b) For real numbers $a,b,c \gt0$ and $n \leq3$ prove that: $$\displaystyle \frac{a}{b}+\frac{b}{c}+\frac{c}{a}+n\left(\frac{3\sqrt[3]{abc}}{a+b+c} \right)\geq 3+n.$$ $\mathbf L\mathbf D\mathbf L^\top$ Cholesky decomposition Improving "demonstrate a minimal understanding" close reason Note: the eventual outcome of this discussion involved dropping this particular off-topic reason, along with other changes. See:  &nbsp; &mdash;Shog     I was using the reason "demonstrate a minimal understanding" myself quite frequently, but the more I review, the more I have doubts about it and I think this reason can either be dropped entirely or at least rephrased, so I started to use "problems with code you've written must describe the specific problem" instead:     Questions asking for code must demonstrate a minimal understanding of the problem being solved.   So the poster has posted some description, maybe even a snippet of code and in his mind that already proves that he has that minimal understanding required. After all he described the problem and what he tried to solve. So telling him that he lacks that understanding is rather rude, and from the posters point of view doesn't even apply. What both version of the reason not emphasize is that a description of a problem is not various versions of     it doesn't work   but that we rather mean really specific details (see below). I can often see in the reopen review queue that the "improved" version just includes a more verbatim description, while what usually would be expected is some specific error message, a trace or a log, or whatever applies.  Then it continues with:     Include attempted solutions, why they didn't work, and the expected results. See also: Stack Overflow question checklist   comparing it to:     problems with code you've written must describe the specific problem — and include valid code to reproduce it — in the question itself. See SSCCE.org for guidance.   Which says essentially the same, only in more detail.   So the "minimum understanding" reason is IMO a more rude version of the reason telling the user that an SSCCE is missing.  The intent of giving this reason is to tell the user that he needs to give exact details to be able to help him, so IMO a checklist would be much better. Something like:     Your question was closed because it may not provide enough details. Things you can do to improve your question to get an answer include:         Format the code in a way that makes it readable and easily understandable.   Provide the smallest working sample of code demonstrating the problem (link to SSCCE)   Include your input and output.   If applicable provide logs (i.e. Android debugging log, stack trace, etc.).   If applicable provide the exact error messages (compiler, etc.).      And something like this should be best included already in the banner that appears when a question is closed instead of being hidden in some pages you often enough have to actively looking for.  It might be helpful to adjust this, based on the tags or close reason, as I realize that I'm more focused on the programming area, and the above list may not apply to other areas of SO.  Update  Another thing I forgot to mention (but wanted to) is this:     Questions asking for code   Is IMO essentially an invitation of the type of questions     I have a problem, can you provide the solution? (with an optional please)   while     Questions concerning problems with code you've written   puts at least some emphasis on the fact that the poster has to put some effort himself in trying to solve the problem by providing the code which he used to attempt the solution. Denoting the end of example/remark What is the etiquette to denote the end of an example or remark? For proofs, it's the QED symbol (white box). I know it's not necessary for everyone, but I would like to use it to break up the text a little. What should I use? A circle or filled box or what? Python idiom to return first item or None try+gerund/infinitive (Difference in meaning) I tried to open the door. I tried opening the door   We can use both infinitive and gerund with verb "try" as used in above sentences. Does it change the meaning of the  sentence if use gerund or infinitive?.(here i don't know if the meaning of sentences is changed.)     The same confusion arise in these two sentences.  1.I stopped seeing him. 2. I stopped to see him.(But in case of these two sentence i know the meaning is changed but don't know exactly what the change is). Book a return flight a year in advance Is it possible to buy an airline ticket that has me returning to my country after a year being abroad? I can't seem to book a flight that far in advance online. Warning: Total photoshop newb!  I recently got a cool image of my fiancee and I holding hands and showing off her engagment ring. The backdrop is the beach:    I tried to trace an image using a combo of the laso and magic wand tools to select my target. Then I used that to pull out the target as another layer.  I then pasted in a background gradient.     Ended up with a really lame photoshoped looking picture. Notice the white lines where I didn't get the arms close enough and the choppy spots where I got too close.    So what's my main problem here?   Is their a way to bet a better cutout? Is their a blending tool that will make the background look more natural? both   I tried a dozen or so tutorials I found in google searches but they just don't blend these types of images together in any of them. Thanks in advance. Troubleshooting "program does not contain a static 'Main' method" when it clearly does...? Proving that kernels of evaluation maps are generated by the $x_i - a_i$ I want to prove that the kernel of the evaluation map $s_a : \mathbb{C}[x_1,\dots,x_n] \rightarrow \mathbb{C}, x_i \mapsto a_i$ where $a = (a_1,\dots, a_n) \in \mathbb{C}^n$ is the ideal generated by $\{x_1 - a_1, \dots, x_n - a_n\}$.    The proof in the book first shows it easily for the case $a = 0$, and says that by substitutions of variables $x_i' = x_i - a_i$, you can show it for the rest of the cases.  I know that $f \in \ker s_a$ iff $f(a) = 0$.  And $\phi : \mathbb{C}[x_1,\dots, x_n] \rightarrow \mathbb{C}[x_1 - a_1, \dots, x_n-a_n], x_i \mapsto x_i - a_i$ is a surjective ring homomorphism.  Please give a hint and not the full answer.  Thanks. I have a service (kado-demo.service) that a normal user (kado) should be able to restart without entering any password.   I created /etc/sudoers.d/user_restart and added  kado ALL=NOPASSWD: /bin/systemctl restart kado-demo.service   Permissions:  -r--r-----   1 root root   585 Mar  5 10:47 user_restart   As user k, I now should be able to execute the given command, but this happens:  /bin/systemctl restart kado-demo.service  ==== AUTHENTICATING FOR org.freedesktop.systemd1.manage-units ===  Authentication is required to restart 'kado-demo.service'.   Multiple identities can be used for authentication:   (...)   Which is the same result as before I added the file.   The /etc/sudoers contains  #includedir /etc/sudoers.d   And I even get the same result when I add content of the user_restart file directly to /etc/sudoers.  I'm running out of ideas... How can I install the Social-Engineering Toolkit (SET)? There is this software (SET) from Backtack 5 source that I found out about. But I don't know how to install this thing. Could somebody help me?  Here the link: Occasionally, when I'm installing stuff, I get an error like the following:  Some packages could not be installed. This may mean that you have requested an impossible situation or if you are using the unstable distribution that some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation:  The following packages have unmet dependencies:  package1 : Depends: package2 (&gt;= 1.8) but 1.7.5-1ubuntu1 is to be installed E: Unable to correct problems, you have held broken packages.   How can I resolve this? The Chinese government has a really specific list of requirements for visa photos:    I had an old unused photo from last time I applied for a visa or passport, and compared it to the requirements (decided not to use it because it was damaged). It seemed to match most of the requirements, except that it seems that they want a photo which isn't square: The required width is less than the required height.  Assuming most drugstores are still providing square passport photos, the question becomes: Should I cut the photo to match the specified width?   What adds to the confusion is if you look at the  the box for the photo is nearly square, not the oblong shape in the requirements. And I have seen other advice that you need to make sure it fully fits the box.  Any advice about what to do here? "Belongs on" comments Earlier today, I was suspended from chat due to an inappropriate message (I totally deserved it, but that's not the point). I looked at my phone, and this is all that it told me:    ...which confused me because chat was still moving as normal. I didn't realize that I had been suspended until I checked on my computer to see what the heck was going on.  To me, "This room is read-only" sounds like some sort of database lockup or something related specifically to the room, not the user. It doesn't give any feedback to indicate that the room will be "read-only" for 30 minutes, and doesn't mention that all other rooms are also "read-only". Even a message saying just "You have been suspended from chat" would be more helpful. Is it acceptable to write a thank you in a comment? There have been a number of cases where I would like to thank someone for providing me with a thorough response and give them feedback beyond the vote up. Often I wish people would tell me why they voted me up. Not that the thank you part is necessary but it sets it apart from the other comments related to answering the question itself.  In this type of situation is a "Thank you" acceptable in a comment?     For more information, see  in the . Could someone help me prove that given a finite measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$, $\displaystyle\lim_{p\to\infty}\|f\|_p=\|f\|_\infty$?    I don't know where to start. GRUB and LILO both fail to install to NVMe hard disk when installing Debian