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If \(f:\mathbb{R}\to(-1,1)\), \(f(x)=\frac{2}{\pi}\tan^{-1}x\), is (f) onto or not?यदि \(f:\mathbb{R}\to(-1,1)\), \(f(x)=\frac{2}{\pi}\tan^{-1}x\), तो (f) सर्वाच्छादक है या नहीं?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: It is ontoसर्वाच्छादक है
Step 1: The range of \(\tan^{-1}x\) is \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\). Step 2: Multiplying by \(\frac{2}{\pi}\) gives the range ((-1,1)), equal to the codomain. Step 3: Carefully apply the effect of a multiplier on the range.चरण 1: \(\tan^{-1}x\) का परास \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\) है। चरण 2: \(\frac{2}{\pi}\) से गुणा करने पर परास ((-1,1)) बनता है, जो सहप्रांत के बराबर है। चरण 3: परास पर गुणक का प्रभाव ध्यान से लगाएँ।
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