Question

Domain of definition of the function $f(x)=\frac{3}{4-x^2}+{\log }_{10}(x^3-x)$ 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

Answer: (D)

$f\left(x\right)=\frac{3}{4-x^2}+lo{g}_{10}\left(x^3-x\right)$
For $D_f,lo{g}_{10}\left(x^3-x\right)>0$
$\Rightarrow x^3-x>0$
$\Rightarrow x\left(x-1\right)\left(x+1\right)>0$

Region is $\left(-1,0\right)\cup \left(1,\infty \right)$ and $4-x^2\ne 0$
$\Rightarrow x\ne 2$
$\therefore$ Region is $\left(-\infty ,2\right)\cup \left(2,\infty \right)$
$\therefore$ Common region is $\left(-1,0\right),\cup \left(1,0\right)\cup \left(2,\infty \right)$