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Fractional Transformation of x

Source: 1971 AHSME Problem 30

April 23, 2014
AMC

Problem Statement

Given the linear fractional transformation of xx into f1(x)=2x−1x+1f_1(x)=\dfrac{2x-1}{x+1}. Define fn+1(x)=f1(fn(x))f_{n+1}(x)=f_1(f_n(x)) for n=1,2,3,⋯n=1,2,3,\cdots. Assuming that f35(x)=f5(x)f_{35}(x)=f_5(x), it follows that f28(x)f_{28}(x) is equal to
<spanclass=′latex−bold′>(A)</span>x<spanclass=′latex−bold′>(B)</span>1x<spanclass=′latex−bold′>(C)</span>x−1x<spanclass=′latex−bold′>(D)</span>11−x<spanclass=′latex−bold′>(E)</span>None of these<span class='latex-bold'>(A) </span>x\qquad<span class='latex-bold'>(B) </span>\frac{1}{x}\qquad<span class='latex-bold'>(C) </span>\frac{x-1}{x}\qquad<span class='latex-bold'>(D) </span>\frac{1}{1-x}\qquad <span class='latex-bold'>(E) </span>\text{None of these}
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