Domain of definition of the function $f(x)=\frac{3}{4-x^2}+{\log }_{10}(x^3-x)$, is:

For $f(x)$ to be defined:The denominator $4-x^2\ne 0⟹x^2\ne 4⟹x\ne \pm 2$.The argument of the log must be positive: $x^3-x>0$.$x(x^2-1)>0⟹x(x-1)(x+1)>0$.Using the wavy curve method for $x(x-1)(x+1)>0$:The intervals are $(-1,0)\cup (1,\infty )$.Combining with the condition $x\ne 2$:The domain is $(-1,0)\cup (1,2)\cup (2,\infty )$.