Official solution
Given \[f(x) = {1 \over {4 - {x^2}}} + {\log _{10}}({x^3} - x)\]
Let f1(x) = \[{1 \over {4 - {x^2}}}\] and f2(x) = \[{\log _{10}}({x^3} - x)\]
Here in f1(x) denominator \[ \ne \] 0
4 - x2 \[ \ne \] 0
\[ \Rightarrow \] x \[ \ne \] \[ \pm \] 2 ...........(1)
and x3 - x > 0
\[ \Rightarrow \] x(x2 - 1) > 0
\[ \Rightarrow \] x(x + 1)(x - 1) > 0
x \[ \in \] (-1, 0) \[ \cup \] (1, \[\infty \]) ........(2)
Hence domain is intersection of (1) and (2)
x \[ \in \] (-1, 0) \[ \cup \] (1, 2) \[ \cup \] (2, \[\infty \])