query: $f(x)f(\frac{1}{x})=1\hspace{1 em}\forall x\in\mathbb{R}$
positive_passages.docid: 29403
positive_passages.text: Finding functions $f: \Bbb R_*^+ \to \Bbb R_*^+$ with certain properties
negative_passages.docid: 90287
negative_passages.text: How to derive $[x_i, F(\vec p)] = i \hbar \frac {\partial F(\vec p)}{\partial p_i}$
negative_passages.docid: 19561
negative_passages.text: Why is $\frac{1}{\frac{1}{X}}=X$?
negative_passages.docid: 51291
negative_passages.text: How can I express $\int \frac{1}{f'(x)}$ in terms of $f(x)$
negative_passages.docid: 50012
negative_passages.text: $f: [a,b] \to \mathbb{R}$, $f(a) = f(b)$. Show $\exists c \in [a, \frac{a+b}{2} ]$ with $f(c)=f(c+\frac{b-a}{2})$
negative_passages.docid: 129579
negative_passages.text: If $f''(x_0)$ exists then $\lim_{x \to x_0} \frac{f(x_0+h)-2f(x_0)+f(x_0-h)}{h^2} = f''(x_0)$
negative_passages.docid: 111756
negative_passages.text: How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?
negative_passages.docid: 52858
negative_passages.text: Find $f$ if $ f(x)+f\left(\frac{1}{1-x}\right)=x $
negative_passages.docid: 132431
negative_passages.text: Proving $|f'(z)|\leq\frac{\text{Re}(f(z))}{\text{Re}(z)}$
negative_passages.docid: 79024
negative_passages.text: the derivation of E(X|X)
negative_passages.docid: 62055
negative_passages.text: Solve $\frac{d}{dx}f(x)=f(x-1)$
negative_passages.docid: 109103
negative_passages.text: Proof that $\lim_{m\to\infty}(1+\frac{r}{m})^{mt}=e^{rt}$
negative_passages.docid: 57695
negative_passages.text: Show $\inf_f\int_0^1|f'(x)-f(x)|dx=1/e$ for continuously differentiable functions with $f(0)=0$, $f(1)=1$.
negative_passages.docid: 132093
negative_passages.text: Solve $f(f(x))=f(x)$
negative_passages.docid: 152841
negative_passages.text: I want to show that $f(x)=x.f(1)$ where $f:R\to R$ is additive.
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: 89125
negative_passages.text: Prove that $r \in R$ such that $0 \lt r \lt 1$ then $\frac{1}{r (1 - r)} \ge 4$