Let $H(x)=3 x^4+6 x^3-2 x^2+1$ and $g(x)$ be a polynomial of degree one. If

$\frac{H(x)}{(x-1)(x+1)(x-2)}=f(x)+\frac{g(x)}{(x-1)(x+1)(x-2)}$ then

$H(-1)+2 H(2)-3 H(1)=$

The number of bijective functions $f: \mathbf{Z} \rightarrow \mathbf{Z}$ such that $f(x+y)=f(x)+f(y) \forall x, y \in \mathbf{Z}$, is

For each $n \in \mathbf{N}$, let $A_n=\{(n+1) k / k \in \mathbf{N}\}$ and $X=\bigcup_{n \in \mathbf{N}} A_n \cdot A$ mapping $f: X \rightarrow N$ defined by $f(x)=x$, $\forall x \in \mathbf{X}$, is

If $f:[-3,2] \rightarrow[0, \sqrt[3]{x}]$ is an onto function defined by $f(n)=\left\{\begin{array}{cc}2+\sqrt[3]{n}, & -3 \leq n \leq-1 \\ n^{2 / 3}, & -1 \leq n \leq 2\end{array}\right.$, then $x=$