# The question is: f(x)=x/x−1, g(x)=1/x, h(x)=x2−1. Find f o g o h and state its domain.

The question is:
$f\left(x\right)=\frac{x}{x-1}$
$g\left(x\right)=\frac{1}{x}$
$h\left(x\right)={x}^{2}-1$
Find $f\circ g\circ h$ and state its domain.
The answer the textbook states is that the domain is all real values of $x$, except $±1$ and $±\sqrt{2}$.
However surely the domain excludes $0$ as well, since $g\left(0\right)$ is undefined.
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Maleah Lester
You're not inputting $x$ into $g$, though. You're inputting $h\left(x\right)$. So yes, $g\left(0\right)$ is undefined, which means that whatever values of $x$ makes $h\left(x\right)=0$ is not part of the domain. That's why they exclude $±1$.

metal1fc
$h\left(x\right)={x}^{2}-1$ has a value for all $R$
$g\left(h\left(x\right)\right)=\frac{1}{{x}^{2}-1}$ which does not have a value at $x=±\sqrt{1}$
$f\left(g\left(h\left(x\right)\right)\right)=\frac{\frac{1}{{x}^{2}-1}}{\frac{1}{{x}^{2}-1}-1}=\frac{1}{1-{x}^{2}+1}=\frac{1}{2-{x}^{2}}$ which does not have a value at $x=±\sqrt{2}$
Hence the total domain is $x\in R-\left\{±\sqrt{1},±\sqrt{2}\right\}$