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Which option correctly describes the domain and range of the function \(f(x)=\frac{1}{x^2+1}\)?फलन \(f(x)=\frac{1}{x^2+1}\) के डोमेन और परास का सही वर्णन कौन-सा है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: Domain \(\mathbb R\), range \((0,1]\)डोमेन \(\mathbb R\), परास \((0,1]\)
Since \(x^2\ge0\), we have \(x^2+1\ge1\), so the denominator never becomes zero and the domain is \(\mathbb R\). Also, \(f(0)=1\) is the maximum value, while the function approaches 0 but never equals it. Exam tip: first check where the denominator is zero.\(x^2\ge 0\) होने से \(x^2+1\ge1\), इसलिए हर वास्तविक \(x\) के लिए हर शून्य नहीं है और डोमेन \(\mathbb R\) है। \(f(0)=1\) अधिकतम है; \(x\) बढ़ने पर मान 0 के निकट जाता है, पर 0 नहीं बनता। परीक्षा टिप: हर के शून्य होने की जाँच करें।
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