query: Iterative functional equation $f \circ f = f$
positive_passages.docid: 59063
positive_passages.text: $f(f(x))=f(x)$ question
negative_passages.docid: 17154
negative_passages.text: Does a non-trivial solution exist for $f'(x)=f(f(x))$?
negative_passages.docid: 32856
negative_passages.text: Functional equations $f(x+y)= f(x) + f(y)$ and $f(xy)= f(x)f(y)$
negative_passages.docid: 10024
negative_passages.text: How to prove $(f \circ\ g) ^{-1} = g^{-1} \circ\ f^{-1}$? (inverse of composition)
negative_passages.docid: 32599
negative_passages.text: Solution to the functional equation $f(x^y)=f(x)^{f(y)}$
negative_passages.docid: 4178
negative_passages.text: Solve $f(x)f(2x^2) = f(2x^3+x)$
negative_passages.docid: 145875
negative_passages.text: Inverse of a bijection f is equal to its derivative
negative_passages.docid: 49211
negative_passages.text: Solving $f(x+y) = f(x)f(y)f(xy)$
negative_passages.docid: 104757
negative_passages.text: If $f \circ g$ is surjective, then f is surjective
negative_passages.docid: 14830
negative_passages.text: A transformation of F[x]
negative_passages.docid: 100319
negative_passages.text: Functional equation with strange property about irrational numbers
negative_passages.docid: 47212
negative_passages.text: Solving functional equation
negative_passages.docid: 19474
negative_passages.text: Find all solutions to the functional equation $f(x) +f(x+y)=y+2 $
negative_passages.docid: 154001
negative_passages.text: $(g\circ f)^{-1}=f^{-1}\circ g^{-1}$
negative_passages.docid: 32432
negative_passages.text: Strategies to find the set of functions $f:\mathbb R\to\mathbb R$ satisfing the functional equation: $f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy))$
negative_passages.docid: 55693
negative_passages.text: The functional equation $ f \big( x - f ( y ) \big) = f \big( f ( y ) \big) + x f ( y ) + f ( x ) - 1 $
negative_passages.docid: 84317
negative_passages.text: Iterated self-composition of arbitrary function