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For which values of \(x\) is the function \(f(x)=\operatorname{sgn}(x^2-4)\) equal to \(0\)?फलन \(f(x)=\operatorname{sgn}(x^2-4)\) का मान किन \(x\) के लिए \(0\) होता है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: \(\{-2,2\}\)\(\{-2,2\}\)
The signum function \(\operatorname{sgn}(t)\) equals \(0\) only when \(t=0\). Hence, solve \(x^2-4=0\), or \((x-2)(x+2)=0\), giving \(x=2\) and \(x=-2\). Option C is incorrect because \(4^2-4\neq0\). Exam tip: To find where \(\operatorname{sgn}(g(x))=0\), set the inner expression \(g(x)\) equal to zero.साइनम फलन \(\operatorname{sgn}(t)\) का मान \(0\) तभी होता है जब \(t=0\) हो। इसलिए \(x^2-4=0\), अर्थात् \((x-2)(x+2)=0\), जिससे \(x=2\) या \(x=-2\) प्राप्त होता है। विकल्प C में \(\pm4\) लेना गलत है, क्योंकि \(4^2-4\neq0\)। परीक्षा-युक्ति: \(\operatorname{sgn}(g(x))=0\) के लिए सीधे \(g(x)=0\) हल करें।
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