Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined as $f(x)=\frac{2 x^2-3 x+2}{3 x^2+x+3}$. Then $f$ is :

Let [ • ] denote the greatest integer function. If the domain of the function $f(x)=\sin ^{-1}\left(\frac{x+[x]}{3}\right)$ is $[\alpha, \beta)$, then $\alpha^2+\beta^2$ is equal to:

For the function $f:[1, \infty) \rightarrow[1, \infty)$ defined by $f(x)=(x-1)^4+1$, among the two statements:

(I) The set $\mathrm{S}=\left\{x \in[1, \infty): f(x)=f^{-1}(x)\right\}$ contains exactly two elements, and

(II) The set $\mathrm{S}=\left\{x \in[1, \infty): f(x)=f^{-1}(x+1)\right\}$ is an empty set,

Let for some $\alpha \in \mathbb{R}, f: \mathbb{R} \rightarrow \mathbb{R}$ be a function satisfying $f(x+y)=f(x)+2 y^2+y+\alpha x y$ for all $x, y \in \mathbb{R}$. If $f(0)=-1$ and $f(1)=2$, then the value of $\sum\limits_{n=1}^5(\alpha+f(n))$ is :