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Which of the following functions has domain all real numbers and range \((0,1]\)?निम्नलिखित में से किस फलन का प्रांत सभी वास्तविक संख्याएँ तथा परास
\((0,1]\) है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: \(f(x)=\frac{1}{x^2+1}\)\(f(x)=\frac{1}{x^2+1}\)
Since \(x^2+1\ge1\), the denominator never becomes zero, so the domain is \(\mathbb R\). Its reciprocal is positive and at most 1, giving range \((0,1]\). Exam tip: check denominator zeros first.\(x^2\ge0\) से \(x^2+1\ge1\), इसलिए हर कभी शून्य नहीं होता और प्रांत \(\mathbb R\) है। इसका व्युत्क्रम धनात्मक तथा अधिकतम 1 है, अतः परास \((0,1]\) है। परीक्षा में पहले हर की शून्यता जाँचें।
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