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If (f:\mathbb{R}\to[0,1]) and (f(x)=\frac{x^2}{1+x^2}), why is (f) not onto?यदि (f:\mathbb{R}\to[0,1]) और (f(x)=\frac{x^2}{1+x^2}), तो (f) सर्वाच्छादक क्यों नहीं है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: Because (1) is never an imageक्योंकि (1) कभी छवि नहीं बनता
Step 1: (\frac{x^2}{1+x^2}<1) for every real (x). Step 2: (0) is obtained at (x=0), but (1) is not obtained though it is in the codomain. Step 3: When a variable occurs in both numerator and denominator, check endpoint values carefully.चरण 1: (\frac{x^2}{1+x^2}<1) हर वास्तविक (x) के लिए है। चरण 2: (x=0) पर (0) मिलता है, पर (1) नहीं मिलता जबकि (1) सहप्रांत में है। चरण 3: जहाँ चर हर और अंश दोनों में हो, सीमा मानों को ध्यान से जाँचें।
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