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Choice A is correct. A tour group with $n$ people, where $n\ge 25$, can be split into two subgroups: the first $25$ people and the additional $n-25$ people. Since the museum charges $\$21$ per person for the first $25$ people and $\$14$ for each additional person, the charge for the first $25$ people is $\$21\left(25\right)$ and the charge for the additional $n-25$ people is $\$14(n-25)$. Therefore, the total charge, in dollars, is given by the function $f\left(n\right)=21\left(25\right)+14(n-25)$, or $f\left(n\right)=14n+175$.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.