question_answer

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

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

B) \[(0,2)\]

C) \[(-1,0)\cup (0,2)\]       

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

Correct Answer: A

Solution :

\[f(x)=\frac{3}{4-{{x}^{2}}}+{{\log }_{10}}({{x}^{3}}-x)\]

\[4-{{x}^{2}}\ne 0;\] \[{{x}^{3}}-x>0;\]

\[x\ne \pm \sqrt{4}\]  and \[-1<x<0\] or \[1<x<\infty \]

\[\therefore \,\,\,D=(-1,0)\cup (1,\infty )-\left\{ \sqrt{4} \right\}\]

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