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If (f:\mathbb{R}\to[0,\infty)) and (f(x)=x^2+|x|), is (f) onto or not?यदि (f:\mathbb{R}\to[0,\infty)) और (f(x)=x^2+|x|), तो (f) सर्वाच्छादक है या नहीं?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: It is ontoसर्वाच्छादक है
Step 1: Put (t=|x|), so (t\ge0) and (f(x)=t^2+t). Step 2: On ([0,\infty)), (t^2+t) takes every value from (0) to (\infty), since (t=\frac{-1+\sqrt{1+4y}}{2}) works. Step 3: For harder modulus questions, substitution is useful.चरण 1: (t=|x|) रखने पर (t\ge0) और (f(x)=t^2+t)। चरण 2: (t^2+t) ([0,\infty)) पर (0) से (\infty) तक सभी मान लेता है, क्योंकि (t=\frac{-1+\sqrt{1+4y}}{2}) मिल जाता है। चरण 3: मापांक वाले कठिन प्रश्नों में नया चर रखना उपयोगी है।
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