<s>[INST] Domain of $f(x)=\log_3\frac{x-1}{x+1}+\sqrt{2x+1} -\frac{1}{x}$ $$f(x)=\log_3\frac{x-1}{x+1}+\sqrt{2x+1} -\frac{1}{x}$$
First:
$\sqrt{2x+1}\ge0$,
$1.$ $x_1\ge-\frac{1}{2}$
$\frac{1}{x}$  , 
$2.$ $x_2\neq0$
We can't have log of negative number so:
$\log_3\frac{x-1}{x+1}\ge 0$
$3.$ $x_3\neq-1$
$x_1$ tells me that $x$ should be bigger then -1/2, let's say if we put x=-2 
 (as -2<-1/2) in the log we get $-3/-1=3$ which is good. I am not sure what is the domain of this function.
 [/INST] $$\log_3\frac{x-1}{x+1}\ge 0$$
This is wrong 
You need $$\frac{x-1}{x+1}> 0  $$ and $$x\ne -1$$ 
Similarly, solve $2x+1\ge 0$ and $x\ne0$ take intersection of all the solutions you get
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