query: If $f(x–y)= f(x)·g (y)–f (y)·g(x)$ and $g(x – y)=g(x)·g(y)+f(x)·f(y)$ for all $x, y \in R$. Find following for the following given conditions.
positive_passages.docid: 156890
positive_passages.text: If $f$ and $g$ satisfy the sine/cosine addition formulae, then what is $g'(0)$?
negative_passages.docid: 129599
negative_passages.text: Functional Equation: Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $(x+y)(f(x)-f(y))=(x-y)f(x+y)$
negative_passages.docid: 45222
negative_passages.text: When $f(x) = g(y)$ for almost every $(x,y)$, must $f$ and $g$ be constant almost everywhere?
negative_passages.docid: 148227
negative_passages.text: Rewriting of functional equation $f(xy)=f(x)+f(y)$
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: 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: 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: 77071
negative_passages.text: Solve the functional equation $f(xy) = f(x)f(y) - f(x + y) + 1$ and $f(1) = 2$
negative_passages.docid: 111556
negative_passages.text: Solving functional equation $f(x+y)+f(x-y)=2f(x)\cos y$?
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: 40609
negative_passages.text: Prove that $f(X)$ and $g(Y)$ are independent if $X$ and $Y$ are independent
negative_passages.docid: 62966
negative_passages.text: Find all functions $f(x)$ such that $f\left(x^2+f(y)\right)$=$y+(f(x))^2$
negative_passages.docid: 118727
negative_passages.text: $g(x+y)+g(x)g(y)=g(xy)+g(x)+g(y)$ for all $x,y$.
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: 95299
negative_passages.text: What can you say about $f$ and $g$ in the case that $fg$ is 1) Injective, 2) Surjective - Cohn - Classic Algebra Page 15
negative_passages.docid: 21891
negative_passages.text: If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t
negative_passages.docid: 409
negative_passages.text: functions satisfying $f(x - y) = f(x) f(y) - f(a - x) f(a + y)$ and $f(0)=1$