જો ગણ \(N\) પ્રાકૃતિક સંખ્યાઓનો ગણ હોય અને બે વિધેયો \(f\) અને \(g\) એ \(f\), \(g : N \to N\) પર \(f\left( n \right) = \left\{ \begin{gathered} \frac{{n + 1}}{2}\,\,\,\,\,\,\,\,\,\,\,{\text{if n is odd}} \hfill \\ \frac{n}{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{if n is even}} \hfill \\
\end{gathered} \right.\) અને \(g(n) = n - (-1)^n\) પર વ્યાખ્યાયિત હોય તો \(fog\) મેળવો.

A વ્યાપ્ત છે અને એક-એક નથી.
B વ્યાપ્ત નથી અને એક-એક છે
C વ્યાપ્ત અને એક-એક બંને છે .
D વ્યાપ્ત અને એક-એક બંને માંથી એકપણ નથી.

Verified Solution

Answer & Solution

Correct Answer

(A) વ્યાપ્ત છે અને એક-એક નથી.

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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