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Domain and range of the function (arctan(ln(sqrtx)-1)))^3

  1. Jan 31, 2012 #1
    1. The problem statement, all variables and given/known data

    f(x) = (arctan(ln(sqrtx - 1)))^3

    2. Relevant equations
    domain of arctan: all real numbers
    range of arctan: -∏/2, ∏/2

    3. The attempt at a solution
    I know that domain is x>0 when x ≠ 1, because I need a positive number to go under the radical and the natural log of 0 is undefined. For the range, then, I worked inwards through the parentheses and then set lnsqrt(x) -1 greater than -pi/2 and less than pi/2. Still, I think I may have made a mistake because my answers keep coming out different. Also, I don't know the effect that the cubed on the whole equation has. Any help to clarify would be greatly appreciated.
  2. jcsd
  3. Jan 31, 2012 #2


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    It's best to work from the inside out.

    In general, the Domain of f(g(x)) is: all values of x in the domain of g, such that g(x) is in the domain of f .

    Finding the range can be a bit trickier.

    Is your function [itex]f(x)=\arctan(\ln(\sqrt{x}-1)\,)\,?[/itex]

    Or is it [itex]f(x)=\arctan(\ln(\sqrt{x-1}\,)\,)\,?[/itex]
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