Let the function f be define as

Question

f(x) = cos(3 arcsin x). Determine f⁽¹⁰⁰⁾(0).

Answer 1

f(x) = cos(3 arcsin 0) = cos(3 * 0) = 1

f(f(x)) = cos(3 arcsin 1) = cos(3 * π/2) =cos(3π/2) = 0

After two steps, the solution returns to 0. This means that, for 2n steps, n an integer, the function returns to 0.

100 = 2(50)

Thus, f¹⁰⁰(0) = 0