Answer
$ (f ◦ g ◦ h)(x) = \frac{1}{(2 − x^2)}$
The domain of $f ◦ g ◦ h$ consists of all $x$ except $±1$ and $±\sqrt{2}$
Work Step by Step
$f(x) = \frac{x}{x - 1}$, $g(x) = \frac{1}{x}$, $h(x) = x^2 − 1$
$g(h(x)) = \frac{1}{x^2 - 1}$
$f(g(h(x))) = \frac{\frac{1}{x^2 - 1}}{(\frac{1}{x^2 - 1}) - 1} = \frac{\frac{1}{x^2 - 1}}{\frac{1 - (x^2 - 1)}{x^2 -1}} = \frac{1}{1- x^2 + 1}$
$f(g(h(x))) = \frac{1}{2 - x^2}$
Domain
For $g(h(x))$ to be defined, we require $h(x) \ne 0$, i.e.
$x \ne ±1$. For $f(g(h(x)))$ to be defined, we also require
$g(h(x)) \ne 1$, i.e. $x \ne ±\sqrt{2}$. So The domain of $f ◦ g ◦ h$ consists of all $x$ except $±1$ and $±\sqrt{2}$.
