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If f(x) = 1/(x² + 1) for x ∈ ℝ, what is the range of f?यदि f(x) = 1/(x² + 1), जहाँ x ∈ ℝ है, तो f का परिसर क्या है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: (0, 1](0, 1]
For every real x, x² ≥ 0, so x² + 1 ≥ 1 and therefore 0 < f(x) ≤ 1. The value 1 is attained at x = 0. The function approaches 0 as |x| grows, but never equals 0, since its numerator is 1. Hence the range is (0, 1].हर वास्तविक x के लिए x² ≥ 0, इसलिए x² + 1 ≥ 1 और 0 < f(x) ≤ 1। x = 0 पर f(x) का मान 1 होता है। |x| बढ़ने पर फलन का मान 0 के निकट पहुँचता है, लेकिन कभी 0 नहीं होता, क्योंकि अंश 1 है। अतः परिसर (0, 1] है।
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