Definition of a unique function

  1. Hi all, I'm wondering whether an expression which is used to describe a function in a certain domain is a different function for the same expression with a differing domain.

    For example: expression; x^2.
    f(x) = x^2 for domain {1 < x < 10}
    f(x) = x^2 for domain {10 < x < 11}

    Are these two f(x)'s the same function, or different functions, by definition. I couldn't be sure by Wikipedia, and it's a difficult question to type into a search engine.
     
  2. jcsd
  3. tiny-tim

    tiny-tim 26,055
    Science Advisor
    Homework Helper

    hi jonsploder! welcome to pf! :smile:
    they're different

    they're both restrictions of the same function defined on the whole of R :wink:
     
  4. Thanks for the welcome, and the reply.
    I know that they are different, however I was wondering, by the most formal definition of a function, whether they are different functions, or if indeed the domain of a function constitutes its identity as a function.
     
  5. tiny-tim

    tiny-tim 26,055
    Science Advisor
    Homework Helper

    they're functions, and they're different

    so they're different functions

    the definition of a function includes its range and domain: different range and/or domain, different functions
     
  6. pasmith

    pasmith 1,200
    Homework Helper

    The domain and codomain are part of the definition of a function.

    Two functions [itex]f : A \to B[/itex] and [itex]g : C \to D[/itex] are equal if and only if [itex]A = C[/itex] and [itex]B = D[/itex] and for all [itex]a \in A[/itex], [itex]f(a) = g(a)[/itex].
     
  7. This definition is exactly correct. That should be your definition.

    ....

    It's worth noting, however, that sometimes people get lazy about codomains and say [itex]f : A \to B[/itex] and [itex]g : C \to D[/itex] are equal when [itex]A = C[/itex] and for all [itex]a \in A[/itex], [itex]f(a) = g(a) \in B\cap D[/itex].
     
  8. pwsnafu

    pwsnafu 902
    Science Advisor

    To explain why this definition is bad, consider
    ##f : \mathbb{R} \to \mathbb{R}##, ##f(x) = 0##
    ##g : \mathbb{R} \to \{0\}##, ##g(x) = 0##.
    Note that under the definition economicnerd gave these would be considered equal. However, g is a surjection while f is not.
     
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