Are there functions with a constant output regardless of input, or functions whose input is limited to a...












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I was wondering whether a function $y=f(x)$ can be defined such that (1) its value $y$ is always constant, no matter what number substitutes $x$, (2) its argument $x$ is limited to a single number? Examples of such function definitions would be greatly appreciated.






functions analytic-geometry







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  • 1




    $begingroup$
    For #2, $f(x) = sqrt{-x^2}$, or $f(x) = sin^{-1} (1+|-xcos x|). There are many functions with limited domains, one just has to mess with the function a bit.
    $endgroup$
    – Christopher Marley
    Jan 28 at 21:43






  • 1




    $begingroup$
    @ChristopherMarley: That is an antediluvian notion of function. (And your second example is 'defined' for all $x=(2n+1)pi$.)
    $endgroup$
    – TonyK
    Jan 28 at 21:47
















0












$begingroup$


I was wondering whether a function $y=f(x)$ can be defined such that (1) its value $y$ is always constant, no matter what number substitutes $x$, (2) its argument $x$ is limited to a single number? Examples of such function definitions would be greatly appreciated.






functions analytic-geometry







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    For #2, $f(x) = sqrt{-x^2}$, or $f(x) = sin^{-1} (1+|-xcos x|). There are many functions with limited domains, one just has to mess with the function a bit.
    $endgroup$
    – Christopher Marley
    Jan 28 at 21:43






  • 1




    $begingroup$
    @ChristopherMarley: That is an antediluvian notion of function. (And your second example is 'defined' for all $x=(2n+1)pi$.)
    $endgroup$
    – TonyK
    Jan 28 at 21:47














0












0








0





$begingroup$


I was wondering whether a function $y=f(x)$ can be defined such that (1) its value $y$ is always constant, no matter what number substitutes $x$, (2) its argument $x$ is limited to a single number? Examples of such function definitions would be greatly appreciated.






functions analytic-geometry







share|cite|improve this question









$endgroup$




I was wondering whether a function $y=f(x)$ can be defined such that (1) its value $y$ is always constant, no matter what number substitutes $x$, (2) its argument $x$ is limited to a single number? Examples of such function definitions would be greatly appreciated.






functions analytic-geometry




functions analytic-geometry






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asked Jan 28 at 21:29









bp2017bp2017

1507




1507








  • 1




    $begingroup$
    For #2, $f(x) = sqrt{-x^2}$, or $f(x) = sin^{-1} (1+|-xcos x|). There are many functions with limited domains, one just has to mess with the function a bit.
    $endgroup$
    – Christopher Marley
    Jan 28 at 21:43






  • 1




    $begingroup$
    @ChristopherMarley: That is an antediluvian notion of function. (And your second example is 'defined' for all $x=(2n+1)pi$.)
    $endgroup$
    – TonyK
    Jan 28 at 21:47














  • 1




    $begingroup$
    For #2, $f(x) = sqrt{-x^2}$, or $f(x) = sin^{-1} (1+|-xcos x|). There are many functions with limited domains, one just has to mess with the function a bit.
    $endgroup$
    – Christopher Marley
    Jan 28 at 21:43






  • 1




    $begingroup$
    @ChristopherMarley: That is an antediluvian notion of function. (And your second example is 'defined' for all $x=(2n+1)pi$.)
    $endgroup$
    – TonyK
    Jan 28 at 21:47








1




1




$begingroup$
For #2, $f(x) = sqrt{-x^2}$, or $f(x) = sin^{-1} (1+|-xcos x|). There are many functions with limited domains, one just has to mess with the function a bit.
$endgroup$
– Christopher Marley
Jan 28 at 21:43




$begingroup$
For #2, $f(x) = sqrt{-x^2}$, or $f(x) = sin^{-1} (1+|-xcos x|). There are many functions with limited domains, one just has to mess with the function a bit.
$endgroup$
– Christopher Marley
Jan 28 at 21:43




1




1




$begingroup$
@ChristopherMarley: That is an antediluvian notion of function. (And your second example is 'defined' for all $x=(2n+1)pi$.)
$endgroup$
– TonyK
Jan 28 at 21:47




$begingroup$
@ChristopherMarley: That is an antediluvian notion of function. (And your second example is 'defined' for all $x=(2n+1)pi$.)
$endgroup$
– TonyK
Jan 28 at 21:47










2 Answers
2






active

oldest

votes


















1












$begingroup$

If $X$ and $Y$ are any sets, then a function from $X$ to $Y$ is simply an assignment of an element of $Y$ to each element of $X$. (More formally, it is a subset of the product set $Xtimes Y$ that satisfies certain conditions.)



So...



For (1), we can define for example a function $f:Bbb RrightarrowBbb R$ by $f(x)=99$ for all $xin Bbb R$.



For (2), we can only define such a function if $X$ consists of a single element. Not very useful!






share|cite|improve this answer









$endgroup$













  • $begingroup$
    What if equation of the function must have input and output variable, $f(x) = ...x...$ or $y = ...x...$ with $...x...$ standing for expression with $x$ variable (which is input)?
    $endgroup$
    – bp2017
    Jan 30 at 17:36








  • 1




    $begingroup$
    @bp2017: that is an obsolete notion of what a function is, and not very useful. See this Wikipedia article for a discussion.
    $endgroup$
    – TonyK
    Jan 30 at 17:46



















1












$begingroup$

A function is two sets with a mapping between them. Its domain could indeed consist of a single element, much like how its output could also be a single number. Think of the identity function $f:{1} to {1}$ where $f(1)=1$. It clearly satisfies both of the properties you give.






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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    1












    $begingroup$

    If $X$ and $Y$ are any sets, then a function from $X$ to $Y$ is simply an assignment of an element of $Y$ to each element of $X$. (More formally, it is a subset of the product set $Xtimes Y$ that satisfies certain conditions.)



    So...



    For (1), we can define for example a function $f:Bbb RrightarrowBbb R$ by $f(x)=99$ for all $xin Bbb R$.



    For (2), we can only define such a function if $X$ consists of a single element. Not very useful!






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      What if equation of the function must have input and output variable, $f(x) = ...x...$ or $y = ...x...$ with $...x...$ standing for expression with $x$ variable (which is input)?
      $endgroup$
      – bp2017
      Jan 30 at 17:36








    • 1




      $begingroup$
      @bp2017: that is an obsolete notion of what a function is, and not very useful. See this Wikipedia article for a discussion.
      $endgroup$
      – TonyK
      Jan 30 at 17:46
















    1












    $begingroup$

    If $X$ and $Y$ are any sets, then a function from $X$ to $Y$ is simply an assignment of an element of $Y$ to each element of $X$. (More formally, it is a subset of the product set $Xtimes Y$ that satisfies certain conditions.)



    So...



    For (1), we can define for example a function $f:Bbb RrightarrowBbb R$ by $f(x)=99$ for all $xin Bbb R$.



    For (2), we can only define such a function if $X$ consists of a single element. Not very useful!






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      What if equation of the function must have input and output variable, $f(x) = ...x...$ or $y = ...x...$ with $...x...$ standing for expression with $x$ variable (which is input)?
      $endgroup$
      – bp2017
      Jan 30 at 17:36








    • 1




      $begingroup$
      @bp2017: that is an obsolete notion of what a function is, and not very useful. See this Wikipedia article for a discussion.
      $endgroup$
      – TonyK
      Jan 30 at 17:46














    1












    1








    1





    $begingroup$

    If $X$ and $Y$ are any sets, then a function from $X$ to $Y$ is simply an assignment of an element of $Y$ to each element of $X$. (More formally, it is a subset of the product set $Xtimes Y$ that satisfies certain conditions.)



    So...



    For (1), we can define for example a function $f:Bbb RrightarrowBbb R$ by $f(x)=99$ for all $xin Bbb R$.



    For (2), we can only define such a function if $X$ consists of a single element. Not very useful!






    share|cite|improve this answer









    $endgroup$



    If $X$ and $Y$ are any sets, then a function from $X$ to $Y$ is simply an assignment of an element of $Y$ to each element of $X$. (More formally, it is a subset of the product set $Xtimes Y$ that satisfies certain conditions.)



    So...



    For (1), we can define for example a function $f:Bbb RrightarrowBbb R$ by $f(x)=99$ for all $xin Bbb R$.



    For (2), we can only define such a function if $X$ consists of a single element. Not very useful!







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Jan 28 at 21:50









    TonyKTonyK

    43.7k358137




    43.7k358137












    • $begingroup$
      What if equation of the function must have input and output variable, $f(x) = ...x...$ or $y = ...x...$ with $...x...$ standing for expression with $x$ variable (which is input)?
      $endgroup$
      – bp2017
      Jan 30 at 17:36








    • 1




      $begingroup$
      @bp2017: that is an obsolete notion of what a function is, and not very useful. See this Wikipedia article for a discussion.
      $endgroup$
      – TonyK
      Jan 30 at 17:46


















    • $begingroup$
      What if equation of the function must have input and output variable, $f(x) = ...x...$ or $y = ...x...$ with $...x...$ standing for expression with $x$ variable (which is input)?
      $endgroup$
      – bp2017
      Jan 30 at 17:36








    • 1




      $begingroup$
      @bp2017: that is an obsolete notion of what a function is, and not very useful. See this Wikipedia article for a discussion.
      $endgroup$
      – TonyK
      Jan 30 at 17:46
















    $begingroup$
    What if equation of the function must have input and output variable, $f(x) = ...x...$ or $y = ...x...$ with $...x...$ standing for expression with $x$ variable (which is input)?
    $endgroup$
    – bp2017
    Jan 30 at 17:36






    $begingroup$
    What if equation of the function must have input and output variable, $f(x) = ...x...$ or $y = ...x...$ with $...x...$ standing for expression with $x$ variable (which is input)?
    $endgroup$
    – bp2017
    Jan 30 at 17:36






    1




    1




    $begingroup$
    @bp2017: that is an obsolete notion of what a function is, and not very useful. See this Wikipedia article for a discussion.
    $endgroup$
    – TonyK
    Jan 30 at 17:46




    $begingroup$
    @bp2017: that is an obsolete notion of what a function is, and not very useful. See this Wikipedia article for a discussion.
    $endgroup$
    – TonyK
    Jan 30 at 17:46











    1












    $begingroup$

    A function is two sets with a mapping between them. Its domain could indeed consist of a single element, much like how its output could also be a single number. Think of the identity function $f:{1} to {1}$ where $f(1)=1$. It clearly satisfies both of the properties you give.






    share|cite|improve this answer











    $endgroup$


















      1












      $begingroup$

      A function is two sets with a mapping between them. Its domain could indeed consist of a single element, much like how its output could also be a single number. Think of the identity function $f:{1} to {1}$ where $f(1)=1$. It clearly satisfies both of the properties you give.






      share|cite|improve this answer











      $endgroup$
















        1












        1








        1





        $begingroup$

        A function is two sets with a mapping between them. Its domain could indeed consist of a single element, much like how its output could also be a single number. Think of the identity function $f:{1} to {1}$ where $f(1)=1$. It clearly satisfies both of the properties you give.






        share|cite|improve this answer











        $endgroup$



        A function is two sets with a mapping between them. Its domain could indeed consist of a single element, much like how its output could also be a single number. Think of the identity function $f:{1} to {1}$ where $f(1)=1$. It clearly satisfies both of the properties you give.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Jan 28 at 21:42

























        answered Jan 28 at 21:36









        Geneten48Geneten48

        1549




        1549






























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