Mathematics - Functions Question with Solution | TestHub

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 $4-x^2\ne 0$

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

The wavy curve for the inequality is

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

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