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Let f(x) = 0 for x < 1 and f(x) = 2x - 2 for x >= 1. How many solutions does the equation f(f(f(x))) = x have?

1. 2
2. 1
3. 3
4. infinitely many

Correct answer: 2

Solution

Solutions of f(f(f(x))) = x include fixed points of f. On the branch x >= 1, f(x) = 2x - 2 has fixed point x = 2, which gives f(f(f(2))) = 2. For x < 1, f(x) = 0, and following the chain f(f(f(x))) can equal x only at x = 0. Checking both branches carefully yields exactly two solutions.

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