I'm solving problems based on composition of functions and stuck in this problem.
If $f(x)=\frac{1}{(x-1)(x-2)}$ and $g(x)=\frac{1}{x^2}$, then find the points of discontinuity of $f(g(x))$.
We know that $f(x)$ is discontinuous at $\{1,2\}$ and $g(x)$ is discontinuous at $\{0\}$
$$f(g(x))=\frac{x^4}{(1-x)(1+x)(1-2x^2)}$$ which seems to be discontinuous at $\{1,-1,\frac{1}{\sqrt{2}}, \frac{-1}{\sqrt{2}}\}$.
Now while writing for the points of discontinuity for $f(g(x))$, do I need to write only $\{1,-1,\frac{1}{\sqrt{2}}, \frac{-1}{\sqrt{2}}\}$ or do I also need to include the discontinuous points of $f(x)$ and $g(x)$ as we must first define $f(x)$ and $g(x)$ properly before finding $f(g(x))$.
We want to find the domain of $f(g(x))$. $ \\ \text{ 1) Domain of } g \text{ is all real numbers but } x=0 \\ \text{ 2) Domain of } f \text{ is all real numbers but } x=1 \text{ or } x=2. \text{ So we need } \frac{1}{x^2} \neq 1 \text{ or } \frac{1}{x^2} \neq 2 \\ \text{ Conclusion: Domain is all real numbers except } x=0 \text{ or } x^2 =1 \text{ or } x^2=\frac{1}{2}$