Range of the function $\mathrm{f}(x)=\frac{x^2+x+2}{x^2+x+1}, x \in \mathbb{R}$ is

Range of the function $\mathrm{f}(x)=\frac{x^2+x+2}{x^2+x+1}, x \in \mathbb{R}$ is
  1. $\left(1, \frac{7}{3}\right)$
  2. $\left[1, \frac{7}{3}\right)$
  3. $\left(1, \frac{7}{3}\right]$
  4. $\left[1, \frac{7}{3}\right]$

Solution

Let $y=\frac{x^2+x+2}{x^2+x+1}$ $\Rightarrow(y-1) x^2+(y-1) x+y-2=0$
For real value of $x, b^2-4 \mathrm{ac} \geq 0$ $\begin{aligned} & \Rightarrow(y-1)^2-4(y-1)(y-2) \geq 0 \\ & \Rightarrow 3 y^2-10 y+7 \leq 0 \\ & \Rightarrow(y-1)(3 y-7) \leq 0 \\ & \Rightarrow 1 \leq y \leq \frac{7}{3} \end{aligned}$
Here, $y \neq 1$ for any $x \in \mathrm{R}$ $\therefore \quad \mathrm{R}_{\mathrm{f}}=\left[1, \frac{7}{3}\right]$

Asked in: MHT CET 2024 (16 May Shift 1)

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