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

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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





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      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


















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