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If \(f:\mathbb{R}\to\mathbb{R}\) is defined by \(f(x)=\frac{x}{1+x^2}\), why is (f) not one-one?यदि \(f:\mathbb{R}\to\mathbb{R}\) को \(f(x)=\frac{x}{1+x^2}\) से परिभाषित किया गया है, तो (f) एकैकी नहीं है। सही कारण कौन सा है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: Because \(f(2)=f\left(\frac{1}{2}\right)\)क्योंकि \(f(2)=f\left(\frac{1}{2}\right)\)
Step 1: \(f(2)=\frac{2}{1+4}=\frac{2}{5}\). Step 2: \(f\left(\frac{1}{2}\right)=\frac{\frac{1}{2}}{1+\frac{1}{4}}=\frac{\frac{1}{2}}{\frac{5}{4}}=\frac{2}{5}\). Step 3: Once two distinct inputs have the same image, the function is not one-one.चरण 1: \(f(2)=\frac{2}{1+4}=\frac{2}{5}\)। चरण 2: \(f\left(\frac{1}{2}\right)=\frac{\frac{1}{2}}{1+\frac{1}{4}}=\frac{\frac{1}{2}}{\frac{5}{4}}=\frac{2}{5}\)। चरण 3: दो अलग निवेशों का समान प्रतिबिंब मिलते ही फलन एकैकी नहीं रहता।
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