Let f 1 ( x ) = f ( x ) , and for n ≥ 1 let f n + 1 ( x ) = f n ( f ( x )) . If f ( x ) = x 3 − 2 3 x 2 + x + 4 1 , find ∫ 0 1 f 2004 ( x ) d x .

Question

Let f 1 ( x ) = f ( x ) , and for n ≥ 1 let f n + 1 ( x ) = f n ( f ( x )) . If f ( x ) = x 3 − 2 3 ​ x 2 + x + 4 1 ​ , find ∫ 0 1 ​ f 2004 ( x ) d x .

TestHub · Accepted Answer

Direct expansion gives f ( 1 − x ) = 1 − f ( x ) . Induction gives f k ( 1 − x ) = 1 − f k ( x ) for positive integers k . Substitution x ↦ 1 − x in the integral then yields J = 1 − J , hence J = 0.50 .

Mathematics - Functions Question with Solution | TestHub

Doubts & Discussion

JEE Advanced 2027 (Class 12+)