If $f : \mathbb{Z} \rightarrow \mathbb{Z}$ is defined by $f(x) = x^{9} - 11 x^{8} - 2 x^{7} + 22 x^{6} + x^{4} - 12 x^{3} + 11 x^{2} + x - 3, \forall x \in \mathbb{Z}$,then $f(11) = $

Let a function $f : R \rightarrow R$ be defined such that $3f(2x^2 - 3x + 5) + 2f(3x^2 - 2x + 4) = x^2 - 7x + 9$ for all $x \in R$. Then the value of $f(5)$ is:

$f(x)$ is a real-valued function such that $2f(x) + 3f(-x) = 15 - 4x$ for all $x \in R$. Then $f(2) =$

How many functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ are there such that $f(x+y)=f(x)+f(y)$ for all $x, y \in \mathbb{Z}$?

Let $f: R \rightarrow R$ be a function defined by $f(x) = \frac{2x+1}{3}$. If $\alpha$ is an element in the domain of $f$ whose image is $\frac{1}{\alpha}$,then the sum of all possible values of such $\alpha$ is

The values of $b$ and $c$ for which the identity $f(x + 1) - f(x) = 8x + 3$ is satisfied,where $f(x) = bx^2 + cx + d$,are: