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Injective function

  1. May 14, 2005 #1
    is a cubic function injective for all real numbers?
  2. jcsd
  3. May 14, 2005 #2


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    Yes it's an injective function.
  4. May 14, 2005 #3


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    depends on your definition of a "cubic function".

    the function f(x) = x^3 + bx^2 + cx + d, could be considred a cubic function.

    it is injective if and only if the derivative f'(x) = 3x^2 + 2bx + c never changes sign.

    e.g. f(x) = x^3 is injective because f'(x) = 3x^2 is never negative.

    but f(x) = x^3 - x is not injective becuase f'(x) = 3x^2 - 1 is positive for x = 1 and negative for x = 0.
  5. May 14, 2005 #4


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    Also because to have an inverse function [itex] f^{-1} (x) [/itex] it must be bijective (injective and surjective).
  6. May 18, 2005 #5
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