f : R → R where $f(x)=\frac{x^2+3x+6}{x^2+x+1}$ then f(x) is

many one and into many one and onto one-one and into one-one and onto

many one and into

Clearly domain of f(x) is all real numbers $f(x)=\frac{-2x^2+10x-3}{(x^2+x+1)^2}=\frac{2(x^2-10x+3)}{(x^2+x+1)^2}$. Now since discriminant of $2x^2 − 10x + 3$ is positive thus f'(x) will not remain its sign throughout the domain. Hence f(x) is many one. Also $\underset{x→±∞}{\lim}f(x)=1$ thus f(x) is into.