Let f: R → R where f(x): (x^2+4x+7)/(x^2+x+1). Is f(x) one-one?

We have $f(x)=\frac{x^2+4x+7}{x^2+x+1}=1+\frac{3(x+2)}{x^2+x+1}$

Let $f(x_1) = f(x_2)$

$⇒1+\frac{3(x_1+2)}{x_1^2+x_1+1}=1+\frac{3(x_2+2)}{x_2^2+x_2+1}$

$⇒x_1x_1^2+x_1x_2+x_1+2x_2^2+2x_2+2$

$x_1^2x_2+x_1x_2+x_2+2x_1^2+2x_1+2$

$⇒(x_1-x_2)(2x_1+2x_2+x_1x_2+1)=0$

Let us consider $2x_1 + 2x_2 +x_1x_2 +1=0$

$⇒x_2=-\frac{1+2x_1}{2+x_1}$

This relation is satisfied by infinite number of pairs $(x_1,x_2)$, where $x_1≠x_2$, e.g., $(0, -1/2), (1, -1)$ etc.

Hence f(x) is many-one.