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Even and odd function question

  1. Feb 1, 2012 #1
    1. The problem statement, all variables and given/known data

    Show that the only function which is both even and odd is [itex]f(x)=0[/itex]

    2. The attempt at a solution

    Since [itex]f(x)=0[/itex] is [itex]f(x)=0x[/itex] it is not hard to show that it is odd and even. In order to complete the proof I need to show that this is the only funcion. I know intuitively that if in [itex]f(x)=Ax[/itex] [itex]A\neq0[/itex] then the function is always either odd, even or neither. How should I complete the proof? What should I write?

    Maybe: "Since, it is obvious that if [itex]A\neq0[/itex] the function is either ... " But is it really that obvious? Should I use proof by exhaustion and show that in all posible cases this is true (when A is positive, negative, fraction)?
     
    Last edited: Feb 1, 2012
  2. jcsd
  3. Feb 1, 2012 #2
    If a function f is both odd and even, then f(-x)=f(x) and f(-x)=-f(x), so...
     
  4. Feb 1, 2012 #3

    tiny-tim

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    hi mindauggas! :smile:

    start "suppose f(x) is not 0 for all x, then …" :wink:
     
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