query: If $f\circ f\circ f=id$, then $f=id$
positive_passages.docid: 143610
positive_passages.text: If $f^3=\rm id$ then it is identity function
negative_passages.docid: 91947
negative_passages.text: Proof of $ f:I_n \rightarrow I_m \Rightarrow n = m$
negative_passages.docid: 101042
negative_passages.text: Suppose we have functions $f:A→B$ and $g:B→C$. Prove that if $f$ and $g$ are invertible, then so is $g \circ f$.
negative_passages.docid: 82597
negative_passages.text: How to find $ f(x)$ if $f(1-f(x))=x$ for all $x$ $\in \mathbb{R}$
negative_passages.docid: 146152
negative_passages.text: Characterising functions $f$ that can be written as $f = g \circ g$?
negative_passages.docid: 47856
negative_passages.text: $f(x)=ax+b$ for some $a,b\in\mathbb{Q}$ if $f(\mathbb{Q})\subset\mathbb{Q}$ and $f(\mathbb{R-Q})\subseteq \mathbb{R}-\mathbb{Q}$
negative_passages.docid: 45710
negative_passages.text: Finishing a proof: $f$ is injective if and only if it has a left inverse
negative_passages.docid: 72694
negative_passages.text: Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
negative_passages.docid: 143354
negative_passages.text: Prove that there exists $c\in\mathbb{R}$ such that $f'(c)=f(c)$
negative_passages.docid: 104369
negative_passages.text: If $f$ continuous and $f(x^2) = f(x)$, then $f$ is a const
negative_passages.docid: 60348
negative_passages.text: How do I prove $F(a)=F(a^2)?$
negative_passages.docid: 6098
negative_passages.text: Let $f:A\rightarrow B$ and $C\subset B,D\subset B$. Prove that $f^{-1}(C\backslash D)=f^{-1}(C)\backslash f^{-1}(D)$
negative_passages.docid: 24466
negative_passages.text: Prove that if $[F(\alpha):F]$ is odd then $F(\alpha)=F(\alpha^2)$
negative_passages.docid: 61640
negative_passages.text: Prove c exists for f:[0,1] -> R
negative_passages.docid: 77074
negative_passages.text: If $ f(f(f(x)))=x$, does$ f(x)=x$ necessarily follow?
negative_passages.docid: 80199
negative_passages.text: Do we have always $f(A \cap B) = f(A) \cap f(B)$?
negative_passages.docid: 76148
negative_passages.text: Show that $\exists c \in \mathbb{R} : f(c) = 0$