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(D) Given the functional equation $f(x) \cdot f(1/x) = f(x) + f(1/x)$.Let $f(x) = 1 + g(x)$,then $(1 + g(x))(1 + g(1/x)) = 1 + g(x) + 1 + g(1/x)$.$1 +

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$(B)$ or $(C)$ both

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(D) Given the functional equation $f(x) \cdot f(1/x) = f(x) + f(1/x)$.Let $f(x) = 1 + g(x)$,then $(1 + g(x))(1 + g(1/x)) = 1 + g(x) + 1 + g(1/x)$.$1 + g(x) + g(1/x) + g(x)g(1/x) = 2 + g(x) + g(1/x)$.$g(x)g(1/x) = 1$,which implies $g(x) = x^n$ or $g(x) = -x^n$.Thus,$f(x) = 1 + x^n$ or $f(x) = 1 - x^n$.Given $f(2) = 9$,we have $1 + 2^n = 9 \Rightarrow 2^n = 8 \Rightarrow n = 3$.So,$f(x) = x^3 + 1$.Now,check the options:$f(1) = 1^3 + 1 = 2$,$f(3) = 3^3 + 1 = 28$. $14 	imes f(1) = 14 	imes 2 = 28 = f(3)$. So $(B)$ is correct.$f(5) = 5^3 + 1 = 126$. $9 	imes f(3) = 9 	imes 28 = 252$. $2 	imes f(5) = 2 	imes 126 = 252$. So $(C)$ is correct.Therefore,both $(B)$ and $(C)$ are correct.

If $f(x)$ is a polynomial function satisfying the condition $f(x) \cdot f(1/x) = f(x) + f(1/x)$ and $f(2) = 9$,then:

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