Question

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

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

Answer 1

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

Given:

$f\left(x\right)=\frac{3}{4-x^2}+{\log }_{10}(x^3-x)$

$\Rightarrow$$1^{st}$ condition: $4-x^2\ne 0$

$\Rightarrow$$x\ne \pm 2$......(1)

$2^{nd}$ condition: $x^3-x>0$

$\Rightarrow$$x(x^2-1)>0$

$\Rightarrow$$x(x-1)(x+1)>0$

$\Rightarrow$$x>1$ or $-1<x<0$.........(2)

Combining the conditions (1) and (2), we get domain as
$(-1,0)\cup (1,2)\cup (2,\infty )$