Take the function $$y=\frac{\sqrt{\cosh\left(\frac{1+x}{x^2}\right) - 1}}{e^{\frac{2}{x-1}\log\left|x-1\right|}+1}$$ I have to find the domain of this function. These are the condition that I set up: $$\begin{cases} e^{\frac{2}{x-1}\log\left|x-1\right|}+1\neq 0&(1)\\ x-1\neq0&(2)\\ \left|x-1\right|>0&(3)\\ \cosh\left(\frac{1+x}{x^2}\right) - 1\ge0&(4)\\ x^2\neq0&(5) \end{cases}$$ And these are the results: $$\begin{cases} \forall x \in\mathbb{R}&(1)\\ x\neq1&(3)\\ \forall x \in\mathbb{R}&(4)\\ x\neq0&(5) \end{cases}$$
$(1)$ Denominator
$(2)$ Denominator of the exponent
$(3)$ Argument of the logarithm
$(4)$ Argument of the root
$(5)$ Denominator of the argument of $\cosh$
And this is the definition set of $y$:
$$x\in(-\infty, 0)\cup(0, 1)\cup(1, +\infty)$$
I deleted $(2)$ because it's included in the $(3)$.
The $(1)$ is verified for all $x$ because it's an exponential and because I set up the $(3)$
To solve $(3)$ I made the $\vee$ between $x-1<0$ and $x-1>0$.
The $(4)$ is verfied for all $x$ because the range of $\cosh(x)$ is $[1;+\infty)$, so it's always greater or equal than $1$.
So, is it correct? Or I was wrong something?
