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Which of the following real-valued functions has all real numbers as its domain and only positive real numbers as its range?निम्नलिखित में से किस वास्तविक-मान फलन का प्रांत सभी वास्तविक संख्याएँ तथा परिसर केवल धनात्मक वास्तविक संख्याएँ हैं?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: \(f(x)=e^x\)\(f(x)=e^x\)
\(e^x\) is defined for every real \(x\) and is always positive. For each \(y>0\), taking \(x=\ln y\) gives \(e^x=y\). \(x^2\) also includes 0 in its range. Exam tip: check domain restrictions in roots and fractions.\(e^x\) प्रत्येक वास्तविक \(x\) के लिए परिभाषित और सदैव धनात्मक है। हर \(y>0\) के लिए \(x=\ln y\) लेने पर \(e^x=y\) मिलता है। \(x^2\) में 0 भी आता है। परीक्षा टिप: वर्गमूल और भिन्न में प्रांत-प्रतिबंध जाँचें।
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