Domain of a function

Discussion in 'Calculus' started by marco.marsella, Jun 24, 2021.

  1. marco.marsella


    Jun 24, 2021
    Likes Received:
    The domain of the function:

    (1) f(x) = x^x

    is known to be [0 .. inf).
    Considering that x = 2x/2, it is possible to rewrite the function as:

    (2) f(x) = x^(2x/2)

    Using the multiplicative property of the exponentials it is possible to rewrite it again as:

    (3) f(x) = (x^2)^(x/2)

    The function (3) is now defined over the whole real axis, as x^2 > 0, for all x.
    The online plotting tools agree with this and draw the functions (1) and (3) on two different domains.
    Since a function cannot have two domains, I suppose I wrote something wrong and the two functions (1) and (3) are indeed different.
    Unfortunately, I can't find what's wrong... the mathematics seems easy...

    Thanks for helping and regards,
    marco.marsella, Jun 24, 2021
  2. marco.marsella


    Jun 27, 2021
    Likes Received:
    there is no need to rewrite given function...domain of f(x)=x^x is all positive values of x:
    {x element R : x>0} (all positive real numbers) (assuming a function from reals to reals) or interval (0,infinity)
    MathLover1, Jun 28, 2021
    nycmathguy likes this.
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