question_answer

Domain of definition of the function \[f(x)=\frac{3}{4-{{x}^{2}}}+{{\log }_{10}}({{x}^{3}}-x)\], is [AIEEE 2003]

A)                    (1, 2)

B)            \[(-1,\ 0)\cup (1,\ 2)\]

C)            \[(1,\ 2)\cup (2,\ \infty )\]

D)            \[(-1,\ 0)\cup (1,\ 2)\cup (2,\ \infty )\]

Correct Answer: D

Solution :

           \[f(x)=\frac{3}{4-{{x}^{2}}}+{{\log }_{10}}({{x}^{3}}-x)\]. So, \[4-{{x}^{2}}\ne 0\]Þ \[x\ne \pm \sqrt{4}\]            and \[{{x}^{3}}-x>0\Rightarrow x({{x}^{2}}-1)>0\] Þ \[x>0,\,x>1\]            \\[D=(-1,\,0)\cup (1,\,\infty )-\{\sqrt{4}\}\]i.e.,\[D=(-1,\,0)\cup (1,\,2)\cup (2,\,\infty )\].