The domain of the definition of the function f (x) = 1 4 - x ^ 2 +...

We know that $\log_{10}\left(g\left(x\right)\right)$ is defined when $g\left(x\right) > 0$ and for a fraction to be defined its denominator must be non-zero.

Thus, the given function is defined only when

${x}^{3} - x > 0$ and $4 - {x}^{2} \ne 0$

$⇒ x\left(x - 1\right)\left(x + 1\right) > 0$ and $x \ne 2, - 2$

The wavy curve for the inequality is

$⇒ x \in \left(- 1, 0\right) \cup \left(1, \infty\right)$ and $x \ne 2, - 2 .$

Hence, domain is \left(- 1, 0\right) \cup \left(1, 2\right) \cup \left(2, \infty\right) .