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If \(f:\mathbb{R}\to\mathbb{R}\) and \(f(x)=\frac{x}{1+x^2}\), what is the correct statement about (f)?यदि \(f:\mathbb{R}\to\mathbb{R}\) तथा \(f(x)=\frac{x}{1+x^2}\) है, तो (f) के बारे में सही कथन क्या है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: (f) is not one-one because \(f(2)=f\left(\frac{1}{2}\right)\)(f) एक-एक नहीं है क्योंकि \(f(2)=f\left(\frac{1}{2}\right)\)
Step 1: For a rational function check repeated values carefully. Step 2: \(f(2)=\frac{2}{5}\) and \(f\left(\frac{1}{2}\right)=\frac{2}{5}\), while \(2\neq\frac{1}{2}\). Step 3: Same value at two different inputs destroys injectivity.चरण 1: भिन्न वाले फलन में समान मानों की जांच करें। चरण 2: \(f(2)=\frac{2}{5}\) और \(f\left(\frac{1}{2}\right)=\frac{2}{5}\), जबकि \(2\neq\frac{1}{2}\)। चरण 3: दो अलग आगतों पर समान मान मिलने से एक-एकता नहीं रहती।
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