Can a non-invertible function be inverted by returning a set of all possible solutions?How should I understand $f^-1(E):=xin A:f(x)in E$?Show that each composite function $f_i circ f_j$ is one of the given functionsThe inverse function of f(x)=ln(x)/x.Find the derivative of $f^-1(x)$ at $x=2$ if $f(x)=x^2 + x + ln x$Inverse function of $x + x^q$ with rational $q$“Class” of functions whose inverse, where defined, is the same “class”Inverse derivative of a functionFinding inverse function of a function with multiple variableTerm for a function that handles all possible inputs?How to show $12^a cdot 18^b$ is injective

Is the infant mortality rate among African-American babies in Youngstown, Ohio greater than that of babies in Iran?

Interview was just a one hour panel. Got an offer the next day; do I accept or is this a red flag?

How could I create a situation in which a PC has to make a saving throw or be forced to pet a dog?

How to ask if I can mow my neighbor's lawn

What kind of chart is this?

Basic power tool set for Home repair and simple projects

Cut power on a remote Raspberry Pi 3 via another raspi

Fill the maze with a wall-following Snake until it gets stuck

Leveraging cash for buying car

First occurrence in the Sixers sequence

Can you cover a cube with copies of this shape?

How did Avada Kedavra get its name?

Should I email my professor to clear up a (possibly very irrelevant) awkward misunderstanding?

Does knowing the surface area of all faces uniquely determine a tetrahedron?

Like a Baby - Riddle

How to search for Android apps without ads?

Co-worker is now managing my team. Does this mean that I'm being demoted?

Catching a robber on one line

Time at 1G acceleration to travel 100000 light years

Numerical second order differentiation

How to make all magic-casting innate, but still rare?

Why is Skinner so awkward in Hot Fuzz?

Print the phrase "And she said, 'But that's his.'" using only the alphabet

1960s sci-fi anthology with a Viking fighting a U.S. army MP on the cover



Can a non-invertible function be inverted by returning a set of all possible solutions?


How should I understand $f^-1(E):=xin A:f(x)in E$?Show that each composite function $f_i circ f_j$ is one of the given functionsThe inverse function of f(x)=ln(x)/x.Find the derivative of $f^-1(x)$ at $x=2$ if $f(x)=x^2 + x + ln x$Inverse function of $x + x^q$ with rational $q$“Class” of functions whose inverse, where defined, is the same “class”Inverse derivative of a functionFinding inverse function of a function with multiple variableTerm for a function that handles all possible inputs?How to show $12^a cdot 18^b$ is injective













2












$begingroup$


Is there a concept of inverting a non-invertible function by returning a set of the possible solutions?



For example:



$g(x) = x^2$



Would it be possible to create an inverse function $f(y)$ where, for example:



$f(4) = -2,2$



(I'm pretty sure I'm going about this wrong, but I'm still learning so I don't know *how* I'm wrong)










share|cite|improve this question









New contributor



Code Slinger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






$endgroup$







  • 3




    $begingroup$
    If you define $f$ as a function from $mathbbR$ to the $textitpower set$ of $mathbbR$, then yes, that would be a function. However, $g circ f$ does not equal the identity anymore, which is a property you'd want/expect from an inverse.
    $endgroup$
    – Alexander Geldhof
    9 hours ago
















2












$begingroup$


Is there a concept of inverting a non-invertible function by returning a set of the possible solutions?



For example:



$g(x) = x^2$



Would it be possible to create an inverse function $f(y)$ where, for example:



$f(4) = -2,2$



(I'm pretty sure I'm going about this wrong, but I'm still learning so I don't know *how* I'm wrong)










share|cite|improve this question









New contributor



Code Slinger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






$endgroup$







  • 3




    $begingroup$
    If you define $f$ as a function from $mathbbR$ to the $textitpower set$ of $mathbbR$, then yes, that would be a function. However, $g circ f$ does not equal the identity anymore, which is a property you'd want/expect from an inverse.
    $endgroup$
    – Alexander Geldhof
    9 hours ago














2












2








2





$begingroup$


Is there a concept of inverting a non-invertible function by returning a set of the possible solutions?



For example:



$g(x) = x^2$



Would it be possible to create an inverse function $f(y)$ where, for example:



$f(4) = -2,2$



(I'm pretty sure I'm going about this wrong, but I'm still learning so I don't know *how* I'm wrong)










share|cite|improve this question









New contributor



Code Slinger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






$endgroup$




Is there a concept of inverting a non-invertible function by returning a set of the possible solutions?



For example:



$g(x) = x^2$



Would it be possible to create an inverse function $f(y)$ where, for example:



$f(4) = -2,2$



(I'm pretty sure I'm going about this wrong, but I'm still learning so I don't know *how* I'm wrong)







functions inverse-function






share|cite|improve this question









New contributor



Code Slinger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.










share|cite|improve this question









New contributor



Code Slinger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.








share|cite|improve this question




share|cite|improve this question








edited 7 hours ago









Asaf Karagila

312k33446780




312k33446780






New contributor



Code Slinger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.








asked 9 hours ago









Code SlingerCode Slinger

1164




1164




New contributor



Code Slinger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.




New contributor




Code Slinger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









  • 3




    $begingroup$
    If you define $f$ as a function from $mathbbR$ to the $textitpower set$ of $mathbbR$, then yes, that would be a function. However, $g circ f$ does not equal the identity anymore, which is a property you'd want/expect from an inverse.
    $endgroup$
    – Alexander Geldhof
    9 hours ago













  • 3




    $begingroup$
    If you define $f$ as a function from $mathbbR$ to the $textitpower set$ of $mathbbR$, then yes, that would be a function. However, $g circ f$ does not equal the identity anymore, which is a property you'd want/expect from an inverse.
    $endgroup$
    – Alexander Geldhof
    9 hours ago








3




3




$begingroup$
If you define $f$ as a function from $mathbbR$ to the $textitpower set$ of $mathbbR$, then yes, that would be a function. However, $g circ f$ does not equal the identity anymore, which is a property you'd want/expect from an inverse.
$endgroup$
– Alexander Geldhof
9 hours ago





$begingroup$
If you define $f$ as a function from $mathbbR$ to the $textitpower set$ of $mathbbR$, then yes, that would be a function. However, $g circ f$ does not equal the identity anymore, which is a property you'd want/expect from an inverse.
$endgroup$
– Alexander Geldhof
9 hours ago











5 Answers
5






active

oldest

votes


















4












$begingroup$

This is a multivalued function (see especially the first example!), or multifunction, or set-valued function. A set-valued map, taking elements of $X$ and producing subsets of $Y$, is often denoted $f : X rightrightarrows Y$.



It can also be denoted more literally by $f : X to 2^Y$, as such maps can be thought of as (ordinary, single-valued) functions from $X$ to the power set of $Y$.



Finally, one could also view them simply as relations with a full domain.






share|cite|improve this answer











$endgroup$












  • $begingroup$
    Multivalued functions were the missing piece - thanks!
    $endgroup$
    – Code Slinger
    5 hours ago


















4












$begingroup$

It is common to use the notation $f^-1(A)$, where $A$ is a subset of the value range of $f$, as a shorthand to describe the set $x : f(x) in A $. Furthermore, if $A$ is a set with only one value $x$ it is also somewhat common to just write $f^-1(x)$ instead of $f^-1(x)$. So if in your case the context is clear, it is fine to write $g^-1(4) = 2, -2$.



Also there are functions that are multivalued by default like the complex logarithm for example.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    So $g^-1(4)$ could be the same as $g^-1(4)$ in a certain context? Would that make $g^-1$ a multivalued function?
    $endgroup$
    – Code Slinger
    5 hours ago


















0












$begingroup$

It is possible to have a set-valued function just like your $f$. The only 'but' is that it is not a function $f: mathbb Rto mathbb R$ but actually $f:mathbb Rtomathcal P(mathbb R)$, i.e. its target is the set of parts (or power set) of $mathbb R$.



What you are actually describing by your $f$ is the level sets of $g$. Any more I would add is contained in @LionCoder's comment, so I invite you to continue reading his input instead of repeating what they have said.






share|cite|improve this answer









$endgroup$




















    0












    $begingroup$

    To give a function is the same to give the sets where the function is constant: $cmapsto f^-1(c)$. This would be a map $F:mathbbRto power(mathbbR)$. The function $F$ is almost injective (since $f^-1(c)cap f^-1(c')=emptyset$ for $cneq c'$). It might not be injective if $f^-1(c)=f^-1(c')=emptyset$, that is, if $c$ and $c'$ are not in the range of $f$. The result is:



    $$F: Image(f)to power(mathbbR)$$



    is injective.



    This map can never be surjective (by cardinality), so this is the best we can say. If you restrict an injective function to its image, then it becomes bijective.






    share|cite|improve this answer









    $endgroup$




















      -1












      $begingroup$

      We usually want functions to only take one value in value space.



      If there are several different possible ones, we call them branches of a function.



      For example $f^-1(t) = -sqrtt$ is one such branch that is inverse to $f(t) = t^2$






      share|cite|improve this answer









      $endgroup$













        Your Answer








        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
        );



        );






        Code Slinger is a new contributor. Be nice, and check out our Code of Conduct.









        draft saved

        draft discarded


















        StackExchange.ready(
        function ()
        StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3262588%2fcan-a-non-invertible-function-be-inverted-by-returning-a-set-of-all-possible-sol%23new-answer', 'question_page');

        );

        Post as a guest















        Required, but never shown

























        5 Answers
        5






        active

        oldest

        votes








        5 Answers
        5






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        4












        $begingroup$

        This is a multivalued function (see especially the first example!), or multifunction, or set-valued function. A set-valued map, taking elements of $X$ and producing subsets of $Y$, is often denoted $f : X rightrightarrows Y$.



        It can also be denoted more literally by $f : X to 2^Y$, as such maps can be thought of as (ordinary, single-valued) functions from $X$ to the power set of $Y$.



        Finally, one could also view them simply as relations with a full domain.






        share|cite|improve this answer











        $endgroup$












        • $begingroup$
          Multivalued functions were the missing piece - thanks!
          $endgroup$
          – Code Slinger
          5 hours ago















        4












        $begingroup$

        This is a multivalued function (see especially the first example!), or multifunction, or set-valued function. A set-valued map, taking elements of $X$ and producing subsets of $Y$, is often denoted $f : X rightrightarrows Y$.



        It can also be denoted more literally by $f : X to 2^Y$, as such maps can be thought of as (ordinary, single-valued) functions from $X$ to the power set of $Y$.



        Finally, one could also view them simply as relations with a full domain.






        share|cite|improve this answer











        $endgroup$












        • $begingroup$
          Multivalued functions were the missing piece - thanks!
          $endgroup$
          – Code Slinger
          5 hours ago













        4












        4








        4





        $begingroup$

        This is a multivalued function (see especially the first example!), or multifunction, or set-valued function. A set-valued map, taking elements of $X$ and producing subsets of $Y$, is often denoted $f : X rightrightarrows Y$.



        It can also be denoted more literally by $f : X to 2^Y$, as such maps can be thought of as (ordinary, single-valued) functions from $X$ to the power set of $Y$.



        Finally, one could also view them simply as relations with a full domain.






        share|cite|improve this answer











        $endgroup$



        This is a multivalued function (see especially the first example!), or multifunction, or set-valued function. A set-valued map, taking elements of $X$ and producing subsets of $Y$, is often denoted $f : X rightrightarrows Y$.



        It can also be denoted more literally by $f : X to 2^Y$, as such maps can be thought of as (ordinary, single-valued) functions from $X$ to the power set of $Y$.



        Finally, one could also view them simply as relations with a full domain.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited 9 hours ago

























        answered 9 hours ago









        Theo BenditTheo Bendit

        23.6k12359




        23.6k12359











        • $begingroup$
          Multivalued functions were the missing piece - thanks!
          $endgroup$
          – Code Slinger
          5 hours ago
















        • $begingroup$
          Multivalued functions were the missing piece - thanks!
          $endgroup$
          – Code Slinger
          5 hours ago















        $begingroup$
        Multivalued functions were the missing piece - thanks!
        $endgroup$
        – Code Slinger
        5 hours ago




        $begingroup$
        Multivalued functions were the missing piece - thanks!
        $endgroup$
        – Code Slinger
        5 hours ago











        4












        $begingroup$

        It is common to use the notation $f^-1(A)$, where $A$ is a subset of the value range of $f$, as a shorthand to describe the set $x : f(x) in A $. Furthermore, if $A$ is a set with only one value $x$ it is also somewhat common to just write $f^-1(x)$ instead of $f^-1(x)$. So if in your case the context is clear, it is fine to write $g^-1(4) = 2, -2$.



        Also there are functions that are multivalued by default like the complex logarithm for example.






        share|cite|improve this answer









        $endgroup$












        • $begingroup$
          So $g^-1(4)$ could be the same as $g^-1(4)$ in a certain context? Would that make $g^-1$ a multivalued function?
          $endgroup$
          – Code Slinger
          5 hours ago















        4












        $begingroup$

        It is common to use the notation $f^-1(A)$, where $A$ is a subset of the value range of $f$, as a shorthand to describe the set $x : f(x) in A $. Furthermore, if $A$ is a set with only one value $x$ it is also somewhat common to just write $f^-1(x)$ instead of $f^-1(x)$. So if in your case the context is clear, it is fine to write $g^-1(4) = 2, -2$.



        Also there are functions that are multivalued by default like the complex logarithm for example.






        share|cite|improve this answer









        $endgroup$












        • $begingroup$
          So $g^-1(4)$ could be the same as $g^-1(4)$ in a certain context? Would that make $g^-1$ a multivalued function?
          $endgroup$
          – Code Slinger
          5 hours ago













        4












        4








        4





        $begingroup$

        It is common to use the notation $f^-1(A)$, where $A$ is a subset of the value range of $f$, as a shorthand to describe the set $x : f(x) in A $. Furthermore, if $A$ is a set with only one value $x$ it is also somewhat common to just write $f^-1(x)$ instead of $f^-1(x)$. So if in your case the context is clear, it is fine to write $g^-1(4) = 2, -2$.



        Also there are functions that are multivalued by default like the complex logarithm for example.






        share|cite|improve this answer









        $endgroup$



        It is common to use the notation $f^-1(A)$, where $A$ is a subset of the value range of $f$, as a shorthand to describe the set $x : f(x) in A $. Furthermore, if $A$ is a set with only one value $x$ it is also somewhat common to just write $f^-1(x)$ instead of $f^-1(x)$. So if in your case the context is clear, it is fine to write $g^-1(4) = 2, -2$.



        Also there are functions that are multivalued by default like the complex logarithm for example.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 9 hours ago









        LionCoderLionCoder

        677315




        677315











        • $begingroup$
          So $g^-1(4)$ could be the same as $g^-1(4)$ in a certain context? Would that make $g^-1$ a multivalued function?
          $endgroup$
          – Code Slinger
          5 hours ago
















        • $begingroup$
          So $g^-1(4)$ could be the same as $g^-1(4)$ in a certain context? Would that make $g^-1$ a multivalued function?
          $endgroup$
          – Code Slinger
          5 hours ago















        $begingroup$
        So $g^-1(4)$ could be the same as $g^-1(4)$ in a certain context? Would that make $g^-1$ a multivalued function?
        $endgroup$
        – Code Slinger
        5 hours ago




        $begingroup$
        So $g^-1(4)$ could be the same as $g^-1(4)$ in a certain context? Would that make $g^-1$ a multivalued function?
        $endgroup$
        – Code Slinger
        5 hours ago











        0












        $begingroup$

        It is possible to have a set-valued function just like your $f$. The only 'but' is that it is not a function $f: mathbb Rto mathbb R$ but actually $f:mathbb Rtomathcal P(mathbb R)$, i.e. its target is the set of parts (or power set) of $mathbb R$.



        What you are actually describing by your $f$ is the level sets of $g$. Any more I would add is contained in @LionCoder's comment, so I invite you to continue reading his input instead of repeating what they have said.






        share|cite|improve this answer









        $endgroup$

















          0












          $begingroup$

          It is possible to have a set-valued function just like your $f$. The only 'but' is that it is not a function $f: mathbb Rto mathbb R$ but actually $f:mathbb Rtomathcal P(mathbb R)$, i.e. its target is the set of parts (or power set) of $mathbb R$.



          What you are actually describing by your $f$ is the level sets of $g$. Any more I would add is contained in @LionCoder's comment, so I invite you to continue reading his input instead of repeating what they have said.






          share|cite|improve this answer









          $endgroup$















            0












            0








            0





            $begingroup$

            It is possible to have a set-valued function just like your $f$. The only 'but' is that it is not a function $f: mathbb Rto mathbb R$ but actually $f:mathbb Rtomathcal P(mathbb R)$, i.e. its target is the set of parts (or power set) of $mathbb R$.



            What you are actually describing by your $f$ is the level sets of $g$. Any more I would add is contained in @LionCoder's comment, so I invite you to continue reading his input instead of repeating what they have said.






            share|cite|improve this answer









            $endgroup$



            It is possible to have a set-valued function just like your $f$. The only 'but' is that it is not a function $f: mathbb Rto mathbb R$ but actually $f:mathbb Rtomathcal P(mathbb R)$, i.e. its target is the set of parts (or power set) of $mathbb R$.



            What you are actually describing by your $f$ is the level sets of $g$. Any more I would add is contained in @LionCoder's comment, so I invite you to continue reading his input instead of repeating what they have said.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 9 hours ago









            Sam SkywalkerSam Skywalker

            54913




            54913





















                0












                $begingroup$

                To give a function is the same to give the sets where the function is constant: $cmapsto f^-1(c)$. This would be a map $F:mathbbRto power(mathbbR)$. The function $F$ is almost injective (since $f^-1(c)cap f^-1(c')=emptyset$ for $cneq c'$). It might not be injective if $f^-1(c)=f^-1(c')=emptyset$, that is, if $c$ and $c'$ are not in the range of $f$. The result is:



                $$F: Image(f)to power(mathbbR)$$



                is injective.



                This map can never be surjective (by cardinality), so this is the best we can say. If you restrict an injective function to its image, then it becomes bijective.






                share|cite|improve this answer









                $endgroup$

















                  0












                  $begingroup$

                  To give a function is the same to give the sets where the function is constant: $cmapsto f^-1(c)$. This would be a map $F:mathbbRto power(mathbbR)$. The function $F$ is almost injective (since $f^-1(c)cap f^-1(c')=emptyset$ for $cneq c'$). It might not be injective if $f^-1(c)=f^-1(c')=emptyset$, that is, if $c$ and $c'$ are not in the range of $f$. The result is:



                  $$F: Image(f)to power(mathbbR)$$



                  is injective.



                  This map can never be surjective (by cardinality), so this is the best we can say. If you restrict an injective function to its image, then it becomes bijective.






                  share|cite|improve this answer









                  $endgroup$















                    0












                    0








                    0





                    $begingroup$

                    To give a function is the same to give the sets where the function is constant: $cmapsto f^-1(c)$. This would be a map $F:mathbbRto power(mathbbR)$. The function $F$ is almost injective (since $f^-1(c)cap f^-1(c')=emptyset$ for $cneq c'$). It might not be injective if $f^-1(c)=f^-1(c')=emptyset$, that is, if $c$ and $c'$ are not in the range of $f$. The result is:



                    $$F: Image(f)to power(mathbbR)$$



                    is injective.



                    This map can never be surjective (by cardinality), so this is the best we can say. If you restrict an injective function to its image, then it becomes bijective.






                    share|cite|improve this answer









                    $endgroup$



                    To give a function is the same to give the sets where the function is constant: $cmapsto f^-1(c)$. This would be a map $F:mathbbRto power(mathbbR)$. The function $F$ is almost injective (since $f^-1(c)cap f^-1(c')=emptyset$ for $cneq c'$). It might not be injective if $f^-1(c)=f^-1(c')=emptyset$, that is, if $c$ and $c'$ are not in the range of $f$. The result is:



                    $$F: Image(f)to power(mathbbR)$$



                    is injective.



                    This map can never be surjective (by cardinality), so this is the best we can say. If you restrict an injective function to its image, then it becomes bijective.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 9 hours ago









                    Hudson LimaHudson Lima

                    664




                    664





















                        -1












                        $begingroup$

                        We usually want functions to only take one value in value space.



                        If there are several different possible ones, we call them branches of a function.



                        For example $f^-1(t) = -sqrtt$ is one such branch that is inverse to $f(t) = t^2$






                        share|cite|improve this answer









                        $endgroup$

















                          -1












                          $begingroup$

                          We usually want functions to only take one value in value space.



                          If there are several different possible ones, we call them branches of a function.



                          For example $f^-1(t) = -sqrtt$ is one such branch that is inverse to $f(t) = t^2$






                          share|cite|improve this answer









                          $endgroup$















                            -1












                            -1








                            -1





                            $begingroup$

                            We usually want functions to only take one value in value space.



                            If there are several different possible ones, we call them branches of a function.



                            For example $f^-1(t) = -sqrtt$ is one such branch that is inverse to $f(t) = t^2$






                            share|cite|improve this answer









                            $endgroup$



                            We usually want functions to only take one value in value space.



                            If there are several different possible ones, we call them branches of a function.



                            For example $f^-1(t) = -sqrtt$ is one such branch that is inverse to $f(t) = t^2$







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered 9 hours ago









                            mathreadlermathreadler

                            16k72263




                            16k72263




















                                Code Slinger is a new contributor. Be nice, and check out our Code of Conduct.









                                draft saved

                                draft discarded


















                                Code Slinger is a new contributor. Be nice, and check out our Code of Conduct.












                                Code Slinger is a new contributor. Be nice, and check out our Code of Conduct.











                                Code Slinger is a new contributor. Be nice, and check out our Code of Conduct.














                                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.




                                draft saved


                                draft discarded














                                StackExchange.ready(
                                function ()
                                StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3262588%2fcan-a-non-invertible-function-be-inverted-by-returning-a-set-of-all-possible-sol%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

                                19. јануар Садржај Догађаји Рођења Смрти Празници и дани сећања Види још Референце Мени за навигацијуу

                                Israel Cuprins Etimologie | Istorie | Geografie | Politică | Demografie | Educație | Economie | Cultură | Note explicative | Note bibliografice | Bibliografie | Legături externe | Meniu de navigaresite web oficialfacebooktweeterGoogle+Instagramcanal YouTubeInstagramtextmodificaremodificarewww.technion.ac.ilnew.huji.ac.ilwww.weizmann.ac.ilwww1.biu.ac.ilenglish.tau.ac.ilwww.haifa.ac.ilin.bgu.ac.ilwww.openu.ac.ilwww.ariel.ac.ilCIA FactbookHarta Israelului"Negotiating Jerusalem," Palestine–Israel JournalThe Schizoid Nature of Modern Hebrew: A Slavic Language in Search of a Semitic Past„Arabic in Israel: an official language and a cultural bridge”„Latest Population Statistics for Israel”„Israel Population”„Tables”„Report for Selected Countries and Subjects”Human Development Report 2016: Human Development for Everyone„Distribution of family income - Gini index”The World FactbookJerusalem Law„Israel”„Israel”„Zionist Leaders: David Ben-Gurion 1886–1973”„The status of Jerusalem”„Analysis: Kadima's big plans”„Israel's Hard-Learned Lessons”„The Legacy of Undefined Borders, Tel Aviv Notes No. 40, 5 iunie 2002”„Israel Journal: A Land Without Borders”„Population”„Israel closes decade with population of 7.5 million”Time Series-DataBank„Selected Statistics on Jerusalem Day 2007 (Hebrew)”Golan belongs to Syria, Druze protestGlobal Survey 2006: Middle East Progress Amid Global Gains in FreedomWHO: Life expectancy in Israel among highest in the worldInternational Monetary Fund, World Economic Outlook Database, April 2011: Nominal GDP list of countries. Data for the year 2010.„Israel's accession to the OECD”Popular Opinion„On the Move”Hosea 12:5„Walking the Bible Timeline”„Palestine: History”„Return to Zion”An invention called 'the Jewish people' – Haaretz – Israel NewsoriginalJewish and Non-Jewish Population of Palestine-Israel (1517–2004)ImmigrationJewishvirtuallibrary.orgChapter One: The Heralders of Zionism„The birth of modern Israel: A scrap of paper that changed history”„League of Nations: The Mandate for Palestine, 24 iulie 1922”The Population of Palestine Prior to 1948originalBackground Paper No. 47 (ST/DPI/SER.A/47)History: Foreign DominationTwo Hundred and Seventh Plenary Meeting„Israel (Labor Zionism)”Population, by Religion and Population GroupThe Suez CrisisAdolf EichmannJustice Ministry Reply to Amnesty International Report„The Interregnum”Israel Ministry of Foreign Affairs – The Palestinian National Covenant- July 1968Research on terrorism: trends, achievements & failuresThe Routledge Atlas of the Arab–Israeli conflict: The Complete History of the Struggle and the Efforts to Resolve It"George Habash, Palestinian Terrorism Tactician, Dies at 82."„1973: Arab states attack Israeli forces”Agranat Commission„Has Israel Annexed East Jerusalem?”original„After 4 Years, Intifada Still Smolders”From the End of the Cold War to 2001originalThe Oslo Accords, 1993Israel-PLO Recognition – Exchange of Letters between PM Rabin and Chairman Arafat – Sept 9- 1993Foundation for Middle East PeaceSources of Population Growth: Total Israeli Population and Settler Population, 1991–2003original„Israel marks Rabin assassination”The Wye River Memorandumoriginal„West Bank barrier route disputed, Israeli missile kills 2”"Permanent Ceasefire to Be Based on Creation Of Buffer Zone Free of Armed Personnel Other than UN, Lebanese Forces"„Hezbollah kills 8 soldiers, kidnaps two in offensive on northern border”„Olmert confirms peace talks with Syria”„Battleground Gaza: Israeli ground forces invade the strip”„IDF begins Gaza troop withdrawal, hours after ending 3-week offensive”„THE LAND: Geography and Climate”„Area of districts, sub-districts, natural regions and lakes”„Israel - Geography”„Makhteshim Country”Israel and the Palestinian Territories„Makhtesh Ramon”„The Living Dead Sea”„Temperatures reach record high in Pakistan”„Climate Extremes In Israel”Israel in figures„Deuteronom”„JNF: 240 million trees planted since 1901”„Vegetation of Israel and Neighboring Countries”Environmental Law in Israel„Executive branch”„Israel's election process explained”„The Electoral System in Israel”„Constitution for Israel”„All 120 incoming Knesset members”„Statul ISRAEL”„The Judiciary: The Court System”„Israel's high court unique in region”„Israel and the International Criminal Court: A Legal Battlefield”„Localities and population, by population group, district, sub-district and natural region”„Israel: Districts, Major Cities, Urban Localities & Metropolitan Areas”„Israel-Egypt Relations: Background & Overview of Peace Treaty”„Solana to Haaretz: New Rules of War Needed for Age of Terror”„Israel's Announcement Regarding Settlements”„United Nations Security Council Resolution 497”„Security Council resolution 478 (1980) on the status of Jerusalem”„Arabs will ask U.N. to seek razing of Israeli wall”„Olmert: Willing to trade land for peace”„Mapping Peace between Syria and Israel”„Egypt: Israel must accept the land-for-peace formula”„Israel: Age structure from 2005 to 2015”„Global, regional, and national disability-adjusted life years (DALYs) for 306 diseases and injuries and healthy life expectancy (HALE) for 188 countries, 1990–2013: quantifying the epidemiological transition”10.1016/S0140-6736(15)61340-X„World Health Statistics 2014”„Life expectancy for Israeli men world's 4th highest”„Family Structure and Well-Being Across Israel's Diverse Population”„Fertility among Jewish and Muslim Women in Israel, by Level of Religiosity, 1979-2009”„Israel leaders in birth rate, but poverty major challenge”„Ethnic Groups”„Israel's population: Over 8.5 million”„Israel - Ethnic groups”„Jews, by country of origin and age”„Minority Communities in Israel: Background & Overview”„Israel”„Language in Israel”„Selected Data from the 2011 Social Survey on Mastery of the Hebrew Language and Usage of Languages”„Religions”„5 facts about Israeli Druze, a unique religious and ethnic group”„Israël”Israel Country Study Guide„Haredi city in Negev – blessing or curse?”„New town Harish harbors hopes of being more than another Pleasantville”„List of localities, in alphabetical order”„Muncitorii români, doriți în Israel”„Prietenia româno-israeliană la nevoie se cunoaște”„The Higher Education System in Israel”„Middle East”„Academic Ranking of World Universities 2016”„Israel”„Israel”„Jewish Nobel Prize Winners”„All Nobel Prizes in Literature”„All Nobel Peace Prizes”„All Prizes in Economic Sciences”„All Nobel Prizes in Chemistry”„List of Fields Medallists”„Sakharov Prize”„Țara care și-a sfidat "destinul" și se bate umăr la umăr cu Silicon Valley”„Apple's R&D center in Israel grew to about 800 employees”„Tim Cook: Apple's Herzliya R&D center second-largest in world”„Lecții de economie de la Israel”„Land use”Israel Investment and Business GuideA Country Study: IsraelCentral Bureau of StatisticsFlorin Diaconu, „Kadima: Flexibilitate și pragmatism, dar nici un compromis în chestiuni vitale", în Revista Institutului Diplomatic Român, anul I, numărul I, semestrul I, 2006, pp. 71-72Florin Diaconu, „Likud: Dreapta israeliană constant opusă retrocedării teritoriilor cureite prin luptă în 1967", în Revista Institutului Diplomatic Român, anul I, numărul I, semestrul I, 2006, pp. 73-74MassadaIsraelul a crescut in 50 de ani cât alte state intr-un mileniuIsrael Government PortalIsraelIsraelIsraelmmmmmXX451232cb118646298(data)4027808-634110000 0004 0372 0767n7900328503691455-bb46-37e3-91d2-cb064a35ffcc1003570400564274ge1294033523775214929302638955X146498911146498911

                                Кастелфранко ди Сопра Становништво Референце Спољашње везе Мени за навигацију43°37′18″ СГШ; 11°33′32″ ИГД / 43.62156° СГШ; 11.55885° ИГД / 43.62156; 11.5588543°37′18″ СГШ; 11°33′32″ ИГД / 43.62156° СГШ; 11.55885° ИГД / 43.62156; 11.558853179688„The GeoNames geographical database”„Istituto Nazionale di Statistica”проширитиууWorldCat156923403n850174324558639-1cb14643287r(подаци)