Let r be a real number and f(x) = 2x-rif x≥rrif x<r.Then, the equation f(x) = f(f(x)) holds for all real values of x where
Explanation:
Case 1: x < r ⇒ f(x) = r ⇒ f(f(x)) = f(f(r)) = 2r – r = r ∴ f(x) = f(f(x))
Case 2: x = r ⇒ f(x) = 2r – r = r ⇒ f(f(x)) = f(f(r)) = 2r – r = r ∴ f(x) = f(f(x))
Case 3: x > r ⇒ f(x) = 2x – r > r ⇒ f(f(x)) = f(f(2x - r)) = 2(2x - r) – r = 4x - 3r ∴ f(x) ≠ f(f(x))
⇒ f(x) = f(f(x) when x ≤ r
Hence, option (d).
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