If $\displaystyle f(x) = \log_{100x} \left(\frac{2\log_{10} x + 2}{-x}\right)$; $\displaystyle g(x) = \{x\}$, 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:

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