I have a function      $$f(x)=\frac{x^2-4}{x-2}$$   while solving this simply I get $f(1)=3,f(2)=\frac00,f(3)=5,\cdots$ the problem is when I solve this simply I get $$f(x)=x+2$$ and then I found except $f(2)$ all the values were same. Can someone help me where I did the mistake?
Forgive my ignorance. The below seems 'inconsistent'. If canceling the $(x+1)$ is 'legal', how does the domain change? I realize it does, but would someone be so kind as to provide an explanation?    $$ \frac{x^2 - 1}{x + 1} \mbox{ is undefined when } x = -1 $$  Its domain (the values that can go into the expression) does not include $ -1 $.  Now, we can factor $ x^2 - 1 $ into $ (x - 1)(x + 1) $ so we get:  $$ \frac{(x - 1)(x + 1)}{(x + 1)} $$  It is now tempting to cancel $ (x + 1) $ from top and bottom to produce:  $$ x - 1 $$  $$ \mbox{Its domain now } \textbf{does} \mbox{ include } -1 \mbox{.} $$  But it is now a different function because it has a different domain.    Thanks!
What is the shortest way to compute the last 3 digits of $17^{256}$? What is the shortest way to compute the last 3 digits of $17^{256}$ ?  My solution: \begin{align} 17^{256} &amp;=289^{128} \\ &amp;=(290 - 1)^{128}\\ &amp;=\binom{128}{0}290^{128} -  ... +\binom{128}{126}290^2 - \binom{128}{127}290 + \binom{128}{128} \end{align}  Computed the last 3 terms whose last 3 digits gave the last 3 digits of $17^{256}$.  Is there any shorter method to do this(which requires much less computation) ? SQL Server Query Plan Hash Collision I'm wondering if anyone can shed light on a worry of mine that a query_plan_hash collision could cause a query to be executed as an entirely different query.  The hash is a 16 digit Hex that sp_help sys.dm_exec_query_stats says is a binary. Therefore it's only a 64 bit hash, and a collision seems very probable (considering SHA1 [160 bit] was just verified to have had collisions).  Would the plan_hash and the query_plan_hash both have to collide for this (query to be executed as an entirely different query) to happen?  I'm also curious if there is a setting somewhere in SQL Server that will allow us to change this hash to SHA2-512 (to reduce the possibility of a collision). Our data is very important.  I searched high and low via Google and Stack Exchange forums. I know plants are green due to chlorophyll.  Surely it would be more beneficial for plants to be red than green as by being green they reflect green light and do not absorb it even though green light has more energy than red light.  Is there no alternative to chlorophyll? Or is it something else? There are various methods to find installed applications on the system but to my knowledge none returns exactly the same list as control panel item Add Remove Programs. Some of the ways I tried are:   WMI via Win32_Product and Win32Reg_AddRemovePrograms Registry, both x32 and x64 branches   For instance, I have 217 entries in AR and   gwmi  Win32_Product   returns 261 items.  gp HKLM:\Software\Microsoft\Windows\CurrentVersion\Uninstall\*, HKLM:\SOFTWARE\Wow6432Node\Microsoft\Windows\CurrentVersion\Uninstall\* | ? {![string]::IsNullOrWhiteSpace($_.DisplayName) } | select DisplayName   returns 119 items.   (gwmi Win32Reg_AddRemovePrograms) + (gwmi Win32Reg_AddRemovePrograms64) | sort DisplayName -Unique    returns 107 items.  I tried to nail it down with more complex filter that included InstallLocation, InstallSource, InstallDate etc without success. Let $R$ be a sub-ring of a ring $S$. Let $f,g$ be non-zero polynomials in $R[x]$ and assume that the leading coefficient of $f$ is a unit in $R$. If $f$ divides $g$ in $S[x]$, show that $f$ divides $g$ in $R[x]$.   Thoughts: If $f$ divides $g$ in $S[x]$, then  $g(x) = f(x)q(x)$ for some $q(x)$ in $S[x]$. From here I'm not sure what the question is asking. Do I have to show that $q(x)$ is an element of $R[x]$ ? Insights appreciated. Concentric circles with CSS C# Mono - Low Level Keyboard Hook How do I increase my code coverage, or why can't I cover these lines? Does the Symmetric difference operator define a group on the powerset of a set? $G$ is the set of all subsets of a set $A$, under the operation of $\triangle\;$: Symmetric Difference of sets.  $A$ has at least two different elements.  I need to check if this is a group, and if it does to show if the group is abelian and/or finite.  Associativity - easy from Symmetric Difference. Identity element - empty group. Inverse element - each element is inverse to itself. Am I right?  abelian? finite? Conditionals in English are usually formed by using if with normal word order; but for the three past (subjunctive) forms were, should, and had, it is also possible to express the conditional through subject–auxiliary inversion alone, with no if in the conditional clause.  Does forming conditionals in this manner differ semantically from if-conditionals? Is some aspect of the conditional statement or the conditionality emphasised more in one version than in the other? Or are there differences in how and when they are used? Or are they simply completely interchangeable?  For example:     If I were you, I would try it again   Were I you, I would try it again      If I had seen it, I would have told you   Had I seen it, I would have told you      If you should drink, don’t get behind the wheel   Should you drink, don’t get behind the wheel Intuitively, why does (B → S) ≡ (¬B ∨ S)? I pursue only intuition; please do not answer with  or . I comprehend the following explanation, but it still does not supply the intuition and I sense a deeper, directer, more intuitive explanation. So how else can I intuit this? Or have I neglected something below? I rewrote: p 51-52.  (2010 2 ed) by Henle, Garfield, Tymoczko.    If the kid misbehaves and the teacher forces him to stay, then the teacher spoke truthfully. Suppose the kid misbehaves but travels. Then clearly the teacher lied. But what if the kid misbehaves, but a big snowstorm forces the trip’s cancellation? You can’t say the teacher lied. The kid behaved but didn’t go; nobody went. The teacher spoke truthfully. 1. If the kid behaved and travels, the teacher spoke truthfully.  Notice that the pattern of Ts and Fs we get, is the pattern for disjunction (see p. 38). How do I read a text file with variable hexadecimal value columns in matlab? Prove $3^n = \sum_{k=0}^n \binom {n} {k} 2^k$ Prove that $f(x)\in \mathbb{Z}[x]$ such that $f(0)$ and $f(1)$ are odd has no integer roots Photoshop CC 2019: Pixelated Brush Stroke For some reasons my brush stroke suddenly looks like this:     As the picture showed, the edges are jagged and pixelated. It doesn't happen before and it's just one of the built-in inking brush.  I know there are a lot of similar questions here but please don't mark this as duplicate. I double checked all the thing the other answers suggested: It's brush tool, not pencil, and my mode of brush and layer is Normal, not Dissolve. It's pixelated even if I draw with mouse so it's not tablet driver issue. I disabled GPU render but nothing changed. How do I fix this? Can a spellcaster use spell slots above their normal limit if their ability scores allow bonus spells of that level? This question is inspired by discussion , about how Heighten Spell works with Reserve Feats.  The question that I am asking here deals with whether or not players can cast spells using spell slots for spell levels that they don't have access to yet.    I feel like an example would explain this concept best.  Take Xylitol, the first-level wizard with an intelligence of 18.  He has 5 0th-level spells per day and 3 1st level spells per day from his class.  He also has 1 bonus 1st level spell per day from his high intelligence.  The  table says that he also has 1 bonus spell of each 2nd, 3rd, and 4th levels.  The question is this:  Can Xylitol cast 1st level spells using the bonus slots for his 2nd, 3rd, and 4th level spells?  He clearly can't cast spells that are inherently higher level than 1st, since he can't learn them, but can he cast 1st level spells using bonus 2nd, 3rd, and 4th level slots?  This has come up on these boards before, .  What I'm looking for here is a clear rules citation that clarifies this one way or the other.  The wording in the section on ability scores makes it quite clear that characters cannot cast spells that are actually higher level than their class allows them to cast, but it is not clear that these higher level slots are barred until the appropriate spell level is unlocked.  If a character has access to a spell level that they can't use because their ability score isn't high enough, they can freely use those slots for lower level spells, so why can't they use their bonus spell slots for lower level spells?  As KRyan's answer says, the Basics section covers this.  Based on questions and comments both here and elsewhere, that section seems to be worded ambiguously enough that it's caused confusion.  I'd like to find another, more clear wording to this rule if at all possible.  Answers from either D&amp;D 3.5 or Pathfinder are acceptable, but please note which one your answer applies to.  N.B. I'd like to note that I personally don't feel that players should have access to higher level slots like this.  There are a large number of locations where it's heavily implied that this is not the case, but I feel that finding a clear rules citation here would be helpful.  I can't find one that is clear for either interpretation here. Which manga chapter correspond to the end of episode 12 of season 2 of the anime? Can a goalkeeper chip the ball to the head of a teammate from a goal kick to circumvent the backpass restriction? How can I display just a portion of an image in HTML/CSS? Problem installing .deb in Software Install Ubuntu 20.04 I tried to install some .deb files in Ubuntu 20.04 from GitHub and other internet sources, but when I open it with Software Install it gives me  problem (Failed to install file: not supported). The only way for me to install them is to go to the downloaded file and do sudo dpkg -i file.deb. Can anyone help? I don't know what to do and I don't want to use the terminal every time. Any help would be appreciated, thanks EDIT: that's NOT a problem with .deb in Ubuntu, it is a problem with Software install. I can install every .deb through the terminal but not with Software install, and Ubuntu suggests to install .deb with Software install. That's not a dependencies problem or anything similar, that's an issue with installing .deb WITH Software install and I would like to know how I can fix Software installer