Question

Check whether $f:R\to R$ be a function defined by $f\left(x\right)=\frac{x^2+2x+5}{x^2+x+1}$ is one-one or not.

Answer 1

Given $f\left(x\right)=\frac{x^2+2x+5}{x^2+x+1}$.

Let us take $f\left(x\right)=4$.

$\frac{x^2+2x+5}{x^2+x+1}=4$

or, $x^2+2x+5=4x^2+4x+4$

or, $3x^2+2x-1=0$

or, $(3x-1)(x+1)=0$

or, $x=\frac{1}{3}$ and $x=-1$.

Now we have for $x=\frac{1}{3}$ and $x=-1$ we have $f\left(x\right)=4$.

So, for two distinct values in domain we have $f\left(x\right)$ has the same image $4$.

$\Rightarrow f\left(x\right)$ can not be one-one.