Let `f: R->R` where, `f(x)=(x^2+4x+7)/(x^2+x+1)` Is `f(x)` one-one?

Explanation

`f(x) = (x^(2)+4x+7)/(x^(2)+x+1)=1+(3(x+2))/(x^(2)+x+1)`
Let `f(x_(1))=f(x_(2))`
`implies 1+(3(x_(1)+2))/(x_(1)^(2)+x_(1)+1)=1+(3(x_(2)+2))/(x_(2)^(2)+x_(2)+1)`
`implies x_(1)x_(2)^(2)+x_(1)x_(2)+x_(1)+2x_(2)^(2)+2x_(2)+2`
`=x_(1)^(2)x_(2)+x_(1)x_(2)+x_(2)+2x_(1)^(2)+2x_(1)+2`
`implies (x_(1)-x_(2))(2x_(1)+2x_(2)+x_(1)x_(2)+1)=0`
Let us consider `2x_(1)+2x_(2)+x_(1)x_(2)+1=0`
`implies x_(2)=(1+2x_(1))/(2+x_(1))`
This relation is satisfied by infinite number of pairs `(x_(1), x_(2))," where " x_(1) ne x_(2), e.g., (0,-1//2),(1,-1) ` etc.
Hence f(x) is many-one.