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Math Help - Find maximum possible range of function for which composition is defined?

  1. #1
    Super Member fardeen_gen's Avatar
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    Find maximum possible range of function for which composition is defined?

    If f(x) = \log_{100x} \left(\frac{2\log_{10} x + 2}{-x}\right); g(x) = \{x\}, where {x} denotes the fractional part of x, find the maximum possible range of g(x) for which the composition f(g(x)) is defined.

    Answer:
    Spoiler:
    R_g = \left(0, \frac{1}{10}\right)


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  2. #2
    MHF Contributor red_dog's Avatar
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    We'll find the domain for f:

    100x>0\Rightarrow x>0

    100x\neq 1\Rightarrow x\neq\frac{1}{100}

    \frac{2\lg x+2}{-x}>0\Rightarrow 2\lg x+2<0\Rightarrow x\in\left(0,\frac{1}{10}\right)

    Then, the maximum domain of f, which must be the maximum range of g, is

    \left(0,\frac{1}{10}\right)-\left\{\frac{1}{100}\right\}
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