Question:
Let
S = {f: ℝ → ℝ ∶ f is a polynomial and f(f(x)) = (f(x))2024 for x ∈ ℝ}.
Then, the number of elements in S is _____________
Let
S = {f: ℝ → ℝ ∶ f is a polynomial and f(f(x)) = (f(x))2024 for x ∈ ℝ}.
Then, the number of elements in S is _____________
S = {f: ℝ → ℝ ∶ f is a polynomial and f(f(x)) = (f(x))2024 for x ∈ ℝ}.
Then, the number of elements in S is _____________
Updated On: Jan 25, 2025
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Correct Answer: 3
Solution and Explanation
The functional equation \( f(f(x)) = f(x)^{2024} \) implies that \( f(x) \) must be a polynomial of a certain form. Upon solving the functional equation, we find that the solutions are limited to constant polynomials, identity functions, and potentially other special cases. After considering all possibilities, the total number of solutions is 3.
Thus, the number of elements in \( S \) is 3.
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