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

1. ## 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.

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

How to do it?

2. 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

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