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Why is \(f:\mathbb{R}\to\mathbb{R}\), where \(f(x)=\tan^{-1}x\), not onto?फलन \(f:\mathbb{R}\to\mathbb{R}\) जहाँ \(f(x)=\tan^{-1}x\), आच्छादी क्यों नहीं है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: Because its range is \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\)क्योंकि इसका परिसर \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\) है
Step 1: Values of \(\tan^{-1}x\) lie only in \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\). Step 2: The codomain is \(\mathbb{R}\), so many real values are missed. Step 3: Remember the principal range of inverse trigonometric functions.चरण 1: \(\tan^{-1}x\) का मान केवल \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\) में आता है। चरण 2: सहप्रांत \(\mathbb{R}\) है इसलिए कई वास्तविक संख्याएँ छूट जाती हैं। चरण 3: प्रतिलोम त्रिकोणमितीय फलनों में मानों की सीमा याद रखें।
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