माना \(N\), प्राकृत संख्याओं का समुच्चय है तथा दो फलन \(f\) तथा \(g\) है, जो \(f, g: N \rightarrow N\) द्वारा इस प्रकार परिभाषित है कि \(f ( n )=\left(\begin{array}{ll}\frac{ n +1}{2} & \text { if }\,\, n \,\,\text { is odd } \\ \frac{ n }{2} & \text { if }\,\, n \,\,\text { is even }\end{array}\right.\) तथा \(g ( n )= n -(-1)^{ n }\) है। तब \(fog\) होगा

A एकैकी तथा आच्छादक दोनों होगा।
B एकैकी परन्तु आच्छादक नहीं होगा।
C ना तो एकैकी ना ही आच्छादक होगा।
D आच्छादक परन्तु एकैकी नहीं होगा।

Verified Solution

Answer & Solution

Correct Answer

(A) एकैकी तथा आच्छादक दोनों होगा।

Step-by-step Solution

Detailed explanation

\(\left. \begin{array}{l} f\left( {g\left( 1 \right)} \right) = 1\\ f\left( {g\left( 2 \right)} \right) = 1 \end{array} \right\}\) Many one \(f\left( {g\left( {2k} \right)} \right) = k\) \(f\left( {g\left( {2k + 1} \right)} \right) = k + 1\) \(\therefore \) Onto

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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