query: Functional equation $f(x+y)=f(x)f(y)-f(xy)+1$
positive_passages.docid: 28286
positive_passages.text: About finding the function such that $f(xy)=f(x)f(y)-f(x+y)+1$
negative_passages.docid: 144090
negative_passages.text: How to solve the functional equation $f\left(x^2+f(y)\right)=y+f(x)^2$
negative_passages.docid: 81055
negative_passages.text: Functional Equation simple problem
negative_passages.docid: 86128
negative_passages.text: Find all functions $ f : \Bbb R \to \Bbb R$ which satisfy $f(x)f(y) = f(x + y) + xy$ for all $x, y ∈ \Bbb R$.
negative_passages.docid: 136475
negative_passages.text: d'Alembert-like functional equation: $f(x+y)+g(x-y)=\lambda f(x)g(y)$
negative_passages.docid: 62744
negative_passages.text: Does the functional equation $f(1/r) = rf(r)$ have any nontrivial solutions besides $f(r) = 1/\sqrt{r}$?
negative_passages.docid: 76458
negative_passages.text: Solve the following functional equation $f(xf(y))+f(yf(x))=2xy$
negative_passages.docid: 41226
negative_passages.text: Find $f:f(xf(x)+f(y))=f(x)^2+y$
negative_passages.docid: 125788
negative_passages.text: A function with a property $f(x+y)=f(x)f(y)$
negative_passages.docid: 69707
negative_passages.text: If $f(x + y) \leq yf(x) + f(f(x))$ for all real numbers $x$ and $y$, then prove that $f(x) = 0$ for all $x ≤ 0$.
negative_passages.docid: 34343
negative_passages.text: Determine all $f : \mathbb R \rightarrow \mathbb R$ such that $f(f(x) + y) = f(x^2 - y) + 4yf(x)$.
negative_passages.docid: 73519
negative_passages.text: Determine all functions satisfying $f(x + f(x + y)) + f(xy) = x + f(x + y) +yf(x)$
negative_passages.docid: 101394
negative_passages.text: Please explain the logic behind $d(xy) = y(dx) + x(dy)$
negative_passages.docid: 81921
negative_passages.text: If $f(x + y) = f(x) + f(y)$ showing that $f(cx) = cf(x)$ holds for rational $c$
negative_passages.docid: 148227
negative_passages.text: Rewriting of functional equation $f(xy)=f(x)+f(y)$
negative_passages.docid: 76679
negative_passages.text: Prove there are no other solutions of functional equation $f(x+y) = \frac{f(x)+f(y)}{1-f(x)f(y)}$
negative_passages.docid: 111789
negative_passages.text: Solving functional differential equation $f'(x)=2f(2x)-f(x)$