Why is union sign not allowed to denote intervals of function increasing?












1














On the calculus exam, we were given a function $$f(x) = frac{2x^2+x-1}{x^2-1}
$$ and asked to find intervals of function increasing and decreasing. In the post-exam solutions PDF, it was written that f(x) is increasing on $$(-infty; -1), (-1; 1), (1; +infty)$$
And specifically noted that writing the function increase interval as $$(-infty; -1) cup (-1; 1) cup (1; +infty)$$ is wrong and unacceptable, however, I fail to understand why so.



So, why is union-sign notation incorrect?






notation







share|cite|improve this question
























  • Increasing on the union requires the function to be increasing across the discontinuities of the domain.
    – Yves Daoust
    21 mins ago
















1














On the calculus exam, we were given a function $$f(x) = frac{2x^2+x-1}{x^2-1}
$$ and asked to find intervals of function increasing and decreasing. In the post-exam solutions PDF, it was written that f(x) is increasing on $$(-infty; -1), (-1; 1), (1; +infty)$$
And specifically noted that writing the function increase interval as $$(-infty; -1) cup (-1; 1) cup (1; +infty)$$ is wrong and unacceptable, however, I fail to understand why so.



So, why is union-sign notation incorrect?






notation







share|cite|improve this question
























  • Increasing on the union requires the function to be increasing across the discontinuities of the domain.
    – Yves Daoust
    21 mins ago














1












1








1







On the calculus exam, we were given a function $$f(x) = frac{2x^2+x-1}{x^2-1}
$$ and asked to find intervals of function increasing and decreasing. In the post-exam solutions PDF, it was written that f(x) is increasing on $$(-infty; -1), (-1; 1), (1; +infty)$$
And specifically noted that writing the function increase interval as $$(-infty; -1) cup (-1; 1) cup (1; +infty)$$ is wrong and unacceptable, however, I fail to understand why so.



So, why is union-sign notation incorrect?






notation







share|cite|improve this question















On the calculus exam, we were given a function $$f(x) = frac{2x^2+x-1}{x^2-1}
$$ and asked to find intervals of function increasing and decreasing. In the post-exam solutions PDF, it was written that f(x) is increasing on $$(-infty; -1), (-1; 1), (1; +infty)$$
And specifically noted that writing the function increase interval as $$(-infty; -1) cup (-1; 1) cup (1; +infty)$$ is wrong and unacceptable, however, I fail to understand why so.



So, why is union-sign notation incorrect?






notation




notation






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 47 mins ago









KM101

3,988417




3,988417










asked 49 mins ago









Rafael Sofi-Zadeh

84




84












  • Increasing on the union requires the function to be increasing across the discontinuities of the domain.
    – Yves Daoust
    21 mins ago


















  • Increasing on the union requires the function to be increasing across the discontinuities of the domain.
    – Yves Daoust
    21 mins ago
















Increasing on the union requires the function to be increasing across the discontinuities of the domain.
– Yves Daoust
21 mins ago




Increasing on the union requires the function to be increasing across the discontinuities of the domain.
– Yves Daoust
21 mins ago










3 Answers
3






active

oldest

votes


















4














The function is decreasing (not increasing, as you said) on each of those three intervals separately. But it is not decreasing on their union because, for example, $f(5)>f(0)$.



Consider a similar case: my bank balance is decreasing from December 9 through December 14, and from December 16 through December 21. Is it therefore decreasing on the entire interval? No, because on December 15 I received my salary, so my bank balance increased sharply between December 14 and 16.






share|cite|improve this answer































    3














    I have never seen one write an interval like $(a;b)$. The standard notation is $(a,b)$.



    Consider a simpler example: $g(x)=frac{-1}{x}$. Note that $g$ is increasing on the interval $(-infty,0)$ and also increasing on the interval $(0,infty)$. However, $g$ is not increasing on the set
    $$
    S=(-infty,0)cup (0,infty).
    $$






    share|cite|improve this answer





























      0














      Because $(-infty,-1)cup(-1,1)cup(1,+infty)$ is not an interval.






      share|cite|improve this answer





















      • For the same equation, for concavity intervals, the answer was given using union-sign notation though. I just don't understand why writing like this would be considered incorrect.
        – Rafael Sofi-Zadeh
        41 mins ago










      • I believe that I provide an answer to that question.
        – José Carlos Santos
        39 mins ago











      Your Answer





      StackExchange.ifUsing("editor", function () {
      return StackExchange.using("mathjaxEditing", function () {
      StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
      StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
      });
      });
      }, "mathjax-editing");

      StackExchange.ready(function() {
      var channelOptions = {
      tags: "".split(" "),
      id: "69"
      };
      initTagRenderer("".split(" "), "".split(" "), channelOptions);

      StackExchange.using("externalEditor", function() {
      // Have to fire editor after snippets, if snippets enabled
      if (StackExchange.settings.snippets.snippetsEnabled) {
      StackExchange.using("snippets", function() {
      createEditor();
      });
      }
      else {
      createEditor();
      }
      });

      function createEditor() {
      StackExchange.prepareEditor({
      heartbeatType: 'answer',
      autoActivateHeartbeat: false,
      convertImagesToLinks: true,
      noModals: true,
      showLowRepImageUploadWarning: true,
      reputationToPostImages: 10,
      bindNavPrevention: true,
      postfix: "",
      imageUploader: {
      brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
      contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
      allowUrls: true
      },
      noCode: true, onDemand: true,
      discardSelector: ".discard-answer"
      ,immediatelyShowMarkdownHelp:true
      });


      }
      });














      draft saved

      draft discarded


















      StackExchange.ready(
      function () {
      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3049628%2fwhy-is-union-sign-not-allowed-to-denote-intervals-of-function-increasing%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      4














      The function is decreasing (not increasing, as you said) on each of those three intervals separately. But it is not decreasing on their union because, for example, $f(5)>f(0)$.



      Consider a similar case: my bank balance is decreasing from December 9 through December 14, and from December 16 through December 21. Is it therefore decreasing on the entire interval? No, because on December 15 I received my salary, so my bank balance increased sharply between December 14 and 16.






      share|cite|improve this answer




























        4














        The function is decreasing (not increasing, as you said) on each of those three intervals separately. But it is not decreasing on their union because, for example, $f(5)>f(0)$.



        Consider a similar case: my bank balance is decreasing from December 9 through December 14, and from December 16 through December 21. Is it therefore decreasing on the entire interval? No, because on December 15 I received my salary, so my bank balance increased sharply between December 14 and 16.






        share|cite|improve this answer


























          4












          4








          4






          The function is decreasing (not increasing, as you said) on each of those three intervals separately. But it is not decreasing on their union because, for example, $f(5)>f(0)$.



          Consider a similar case: my bank balance is decreasing from December 9 through December 14, and from December 16 through December 21. Is it therefore decreasing on the entire interval? No, because on December 15 I received my salary, so my bank balance increased sharply between December 14 and 16.






          share|cite|improve this answer














          The function is decreasing (not increasing, as you said) on each of those three intervals separately. But it is not decreasing on their union because, for example, $f(5)>f(0)$.



          Consider a similar case: my bank balance is decreasing from December 9 through December 14, and from December 16 through December 21. Is it therefore decreasing on the entire interval? No, because on December 15 I received my salary, so my bank balance increased sharply between December 14 and 16.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 17 mins ago


























          community wiki





          2 revs
          MJD
























              3














              I have never seen one write an interval like $(a;b)$. The standard notation is $(a,b)$.



              Consider a simpler example: $g(x)=frac{-1}{x}$. Note that $g$ is increasing on the interval $(-infty,0)$ and also increasing on the interval $(0,infty)$. However, $g$ is not increasing on the set
              $$
              S=(-infty,0)cup (0,infty).
              $$






              share|cite|improve this answer


























                3














                I have never seen one write an interval like $(a;b)$. The standard notation is $(a,b)$.



                Consider a simpler example: $g(x)=frac{-1}{x}$. Note that $g$ is increasing on the interval $(-infty,0)$ and also increasing on the interval $(0,infty)$. However, $g$ is not increasing on the set
                $$
                S=(-infty,0)cup (0,infty).
                $$






                share|cite|improve this answer
























                  3












                  3








                  3






                  I have never seen one write an interval like $(a;b)$. The standard notation is $(a,b)$.



                  Consider a simpler example: $g(x)=frac{-1}{x}$. Note that $g$ is increasing on the interval $(-infty,0)$ and also increasing on the interval $(0,infty)$. However, $g$ is not increasing on the set
                  $$
                  S=(-infty,0)cup (0,infty).
                  $$






                  share|cite|improve this answer












                  I have never seen one write an interval like $(a;b)$. The standard notation is $(a,b)$.



                  Consider a simpler example: $g(x)=frac{-1}{x}$. Note that $g$ is increasing on the interval $(-infty,0)$ and also increasing on the interval $(0,infty)$. However, $g$ is not increasing on the set
                  $$
                  S=(-infty,0)cup (0,infty).
                  $$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 24 mins ago









                  user587192

                  1,518112




                  1,518112























                      0














                      Because $(-infty,-1)cup(-1,1)cup(1,+infty)$ is not an interval.






                      share|cite|improve this answer





















                      • For the same equation, for concavity intervals, the answer was given using union-sign notation though. I just don't understand why writing like this would be considered incorrect.
                        – Rafael Sofi-Zadeh
                        41 mins ago










                      • I believe that I provide an answer to that question.
                        – José Carlos Santos
                        39 mins ago
















                      0














                      Because $(-infty,-1)cup(-1,1)cup(1,+infty)$ is not an interval.






                      share|cite|improve this answer





















                      • For the same equation, for concavity intervals, the answer was given using union-sign notation though. I just don't understand why writing like this would be considered incorrect.
                        – Rafael Sofi-Zadeh
                        41 mins ago










                      • I believe that I provide an answer to that question.
                        – José Carlos Santos
                        39 mins ago














                      0












                      0








                      0






                      Because $(-infty,-1)cup(-1,1)cup(1,+infty)$ is not an interval.






                      share|cite|improve this answer












                      Because $(-infty,-1)cup(-1,1)cup(1,+infty)$ is not an interval.







                      share|cite|improve this answer












                      share|cite|improve this answer



                      share|cite|improve this answer










                      answered 47 mins ago









                      José Carlos Santos

                      148k22117218




                      148k22117218












                      • For the same equation, for concavity intervals, the answer was given using union-sign notation though. I just don't understand why writing like this would be considered incorrect.
                        – Rafael Sofi-Zadeh
                        41 mins ago










                      • I believe that I provide an answer to that question.
                        – José Carlos Santos
                        39 mins ago


















                      • For the same equation, for concavity intervals, the answer was given using union-sign notation though. I just don't understand why writing like this would be considered incorrect.
                        – Rafael Sofi-Zadeh
                        41 mins ago










                      • I believe that I provide an answer to that question.
                        – José Carlos Santos
                        39 mins ago
















                      For the same equation, for concavity intervals, the answer was given using union-sign notation though. I just don't understand why writing like this would be considered incorrect.
                      – Rafael Sofi-Zadeh
                      41 mins ago




                      For the same equation, for concavity intervals, the answer was given using union-sign notation though. I just don't understand why writing like this would be considered incorrect.
                      – Rafael Sofi-Zadeh
                      41 mins ago












                      I believe that I provide an answer to that question.
                      – José Carlos Santos
                      39 mins ago




                      I believe that I provide an answer to that question.
                      – José Carlos Santos
                      39 mins ago


















                      draft saved

                      draft discarded




















































                      Thanks for contributing an answer to Mathematics Stack Exchange!


                      • Please be sure to answer the question. Provide details and share your research!

                      But avoid …



                      • Asking for help, clarification, or responding to other answers.

                      • Making statements based on opinion; back them up with references or personal experience.


                      Use MathJax to format equations. MathJax reference.


                      To learn more, see our tips on writing great answers.





                      Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


                      Please pay close attention to the following guidance:


                      • Please be sure to answer the question. Provide details and share your research!

                      But avoid …



                      • Asking for help, clarification, or responding to other answers.

                      • Making statements based on opinion; back them up with references or personal experience.


                      To learn more, see our tips on writing great answers.




                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function () {
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3049628%2fwhy-is-union-sign-not-allowed-to-denote-intervals-of-function-increasing%23new-answer', 'question_page');
                      }
                      );

                      Post as a guest















                      Required, but never shown





















































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown

































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown







                      -

                      Popular posts from this blog

                      Contact image not getting when fetch all contact list from iPhone by CNContact

                      Image
                      up vote 0 down vote favorite I know this question already asked but not getting solution. From this code I will get all the information from the contact but image not found when open vcf files on mac os, also not getting when share this file. I use this stackoverflow link here but It's not help full. var contacts = [CNContact]() let keys = [CNContactVCardSerialization.descriptorForRequiredKeys() ] as [Any] let request = CNContactFetchRequest(keysToFetch: keys as! [CNKeyDescriptor]) do { try self.contactStore.enumerateContacts(with: request) { (contact, stop) in // Array containing all unified contacts from everywhere contacts.append(contact) } } catch { print("unable to fetch contacts") } do { let data = try
                      Read more

                      count number of partitions of a set with n elements into k subsets

                      Image
                      up vote 6 down vote favorite 1 This program is for count number of partitions of a set with n elements into k subsets I am confusing here return k*countP(n-1, k) + countP(n-1, k-1); can some one explain what is happening here? why we are multiplying with k? NOTE->I know this is not the best way to calculate number of partitions that would be DP // A C++ program to count number of partitions // of a set with n elements into k subsets #include<iostream> using namespace std; // Returns count of different partitions of n // elements in k subsets int countP(int n, int k) { // Base cases if (n == 0 || k == 0 || k > n) return 0; if (k == 1 || k == n) return 1; // S(n+1, k) = k*S(n, k) + S(n, k-1) return k*countP(n-1, k) + countP(n-1, k-1); } // Driver program int ma
                      Read more

                      A CLEAN and SIMPLE way to add appendices to Table of Contents and bookmarks

                      Image
                      0 I found a lot of questions abount appendices and ToC. Many users want appendices to be grouped in an Appendix part, however some problems arise with ToC, hyperref, PDF viewer bookmarks, and so on. There are different solutions which require extra packages, command patching and other extra code, however none of them satisfies me. I almost found an easy way to accomplish a good result, where appendices are added to bookmarks in the right way and hyperref links point to the right page. However, the number of the "Appendix" part page is wrong (it's the number of appendix A). Is there any EASY way to fix that? This is a MWE: documentclass{book} usepackage[nottoc,notlot,notlof]{tocbibind} usepackage{hyperref} begin{document} frontmatter tableofcontents mainmatter part{First} chapter{
                      Read more