Question

What is the domain of $f\left(x\right)=\frac{3}{4-x^2}+{\log }_{10}(x^3-x)$?

Answer 1

Let $f\left(x\right)=\frac{3}{4-x^2}+{\log }_{10}(x^3-x)$

For the functions to be defined:

$4-x^2\ne 0\Rightarrow x^2\ne 4$

$\Rightarrow x\ne \pm 2$

Also $x^3-x>0$

$\Rightarrow x(x^2-1)>0\Rightarrow x(x-1)(x+1)>0\Rightarrow x\in (-1,0)\cup (1,\infty )$
So, the domain of function is $x\in (-1,0)\cup (1,\infty )-\left\{2\right\}$.