f-n-0-1-0-1-g-0-1-0-1-f-n-1-x-g-f-n-x-f-n-g-x-f-0-x-x-f-4-x-g-x-x-2-f-2-2-

Question Number 2253 by 123456 last updated on 11/Nov/15

f_{n} : [0, 1] \to [0, 1], g : [0, 1] \to [0, 1]

f_{n + 1} (x) = g [f_{n} (x)] + f_{n} [g (x)]

f_{0} (x) = x

f_{4} (x) = ?

g (x) = x^{2}, f_{2} (2) = ? ?

Commented by RasheedAhmad last updated on 12/Nov/15

D e f i n i t i o n g (x) = x^{2} o n l y f o r f_{2} (2) ?

Answered by prakash jain last updated on 12/Nov/15

f_{1} (x) = g [f_{0} (x)] + f_{0} [g (x)] = g (x) + g (x) = 2 g (x)

f_{1} (x) = 2 x^{2}

f_{2} (x) = g (2 x^{2}) + f_{1} (x^{2}) = 4 x^{4} + 2 x^{4} = 6 x^{4}

f_{3} (x) = {(6 x^{4})}^{2} + 6 x^{8} = 42 x^{8}

f_{4} (x) = {(42 x^{8})}^{2} + 42 {(x^{2})}^{8} = 42 \times 43 x^{16}