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If \(f:\mathbb{R}\to\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\), where \(f(x)=\tan^{-1}x\), what is the correct conclusion?यदि \(f:\mathbb{R}\to\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\) जहाँ \(f(x)=\tan^{-1}x\), तो सही निष्कर्ष क्या है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: (f) is one-one and onto(f) एकैकी और आच्छादी है
Step 1: \(\tan^{-1}x\) is strictly increasing, so it is one-one. Step 2: Its range is \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\), equal to the codomain. Step 3: Choosing the exact range as codomain often makes a function onto.चरण 1: \(\tan^{-1}x\) सख्ती से बढ़ता है इसलिए एकैकी है। चरण 2: इसका परिसर \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\) है जो सहप्रांत के बराबर है। चरण 3: सही सहप्रांत चुनने पर कई फलन आच्छादी बन जाते हैं।
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