1978 AHSME Problem 17

If k is a positive number and f is a function such that, for every positive number x, \left[f(x^2+1)\right]^{\sqrt{x}}=k; then, for every positive number y, \left[f(\frac{9+y^2}{y^2})\right]^{\sqrt{\frac{12}{y}}} is equal to

\textbf{(A) }\sqrt{k}\qquad \textbf{(B) }2k\qquad \textbf{(C) }k\sqrt{k}\qquad \textbf{(D) }k^2\qquad \textbf{(E) }y\sqrt{k}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.