Question

The domain of the definition of the function $f\left(x\right)=\frac{1}{4-x^2}+{\log }_{10}(x^3-x)$ is :

• A. $(-1,0)\cup (1,2)\cup (3,\infty )$
• B. $(1,2)\cup (2,\infty )$
• C. $(-1,0)\cup (1,2)\cup (2,\infty )$
• D. $(-2,-1)\cup (-1,0)\cup (2,\infty )$

Answer 1

The correct option is C $(-1,0)\cup (1,2)\cup (2,\infty )$

$f\left(x\right)=\frac{1}{4-x^2}+{\log }_{10}(x^3-x)$
For $f\left(x\right)$ to be defined,
$4-x^2\ne 0$ and $x^3-x>0$
$\Rightarrow x\ne \pm 2$ and $x(x-1)(x+1)>0$
$\Rightarrow x\in (-1,0)\cup (1,2)\cup (2,\infty )$