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)

$f\left(x\right) = \frac{3}{4-x^{2}} + log_{10}\left(x^{3}-x\right)$
For $D_{f}, log_{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 $4-x^{2}\ne 0$
$\Rightarrow x \ne2$
$ \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) $