2. Write the domain of the function $f(x)=\log _{\frac{1}{2}}\left(\frac{x}{x+2}-\frac{1}{x}\right)-\frac{1}{2}$.
2. We find that the argument of the logarithm $\frac{x}{x+2}-\frac{1}{x}$ must be greater than 0. We write the inequality $\frac{x}{x+2}-\frac{1}{x}>0$. We simplify the inequality $\frac{x^{2}-x-2}{x(x+2)}>0$. We determine the roots $x_{1}=2$ and $x_{2}=-1$ and the poles $x_{1}=0$ and $x_{2}=-2$. On the number line, we determine the signs and read off the solution $x \in(-\infty,-2) \cup(-1,0) \cup(2, \infty)$.

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![](https://cdn.mathpix.com/cropped/2024_06_07_6d1b766ff5bddd57ba61g-13.jpg?height=63&width=1639&top_left_y=1119&top_left_x=208)

![](https://cdn.mathpix.com/cropped/2024_06_07_6d1b766ff5bddd57ba61g-13.jpg?height=48&width=1636&top_left_y=1181&top_left_x=210)

![](https://cdn.mathpix.com/cropped/2024_06_07_6d1b766ff5bddd57ba61g-13.jpg?height=49&width=1636&top_left_y=1226&top_left_x=210)

The signs marked on the number line or the drawn graph................................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_6d1b766ff5bddd57ba61g-13.jpg?height=51&width=1636&top_left_y=1322&top_left_x=210)