query: If $\space$ $\forall$ $x \in \Bbb R$, $\space$ $f(f(x))=x^2-x+1$. Find the value of $f(0)$.
positive_passages.docid: 151519
positive_passages.text: If $f(f(x))=x^2-x+1$, what is $f(0)$?
negative_passages.docid: 44129
negative_passages.text: If $f:\mathbb{R}\rightarrow \mathbb{R}$ is differentiable and $\lim_{x\to \infty } f^\prime(x)=0$ show that $\lim _{x\to \infty } (f(x+1)-f(x))=0$.
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: 35556
negative_passages.text: $f(z)=\frac{z}{1+|z|} $ Is 1-1?
negative_passages.docid: 30860
negative_passages.text: Prove that $f(x)=0$ in $[-1,1]$.
negative_passages.docid: 133646
negative_passages.text: real analysis -functional equation
negative_passages.docid: 26207
negative_passages.text: If $f$ is a quadratic and $f(x)>0\;\forall x$, and $g= f + f' + f''$, prove $g(x)>0\; \forall x$
negative_passages.docid: 120774
negative_passages.text: Show that $f$ has exactly one zero on the square $Q =$ {$x + iy ∈ \Bbb C : |x| < 1, |y| < 1$}.
negative_passages.docid: 98255
negative_passages.text: Find $f(x)$ such that $f(f(x)) = x^2 - 2$
negative_passages.docid: 64339
negative_passages.text: Find $f$ where $f'(x) = f(1+x)$
negative_passages.docid: 150502
negative_passages.text: If $f(0) = 0$ and $|f'(x)|\leq |f(x)|$ for all $x\in\mathbb{R}$ then $f\equiv 0$
negative_passages.docid: 129902
negative_passages.text: If $f(x)=x+\int_{0}^{1}[xy^2+x^2y]f(y)dy$ where $x$ and$y$ are independent variable.Find $f(x).$
negative_passages.docid: 21954
negative_passages.text: A question about $f(x)\equiv 0$
negative_passages.docid: 149328
negative_passages.text: A function such that $f:\mathbb C \to \mathbb C$ such that $f(z)f(iz)=z^2$ satisfies $f(z)+f(-z)=0$.(MADHAVA-2020)
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: 69869
negative_passages.text: if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
negative_passages.docid: 65946
negative_passages.text: A function $f: \mathbb R \to \mathbb R$ satisfies equation $f(x+y)=f(x)f(y)$ for all $x,y\in \mathbb R $